In order to find the extrema, we need to solve the equation
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
$$\frac{d}{d x} f{\left(x \right)} = $$
the first derivative$$- \frac{12 x^{3} \left(3 x + \left(x^{4} + 1\right)\right)}{\left(3 x^{4} + 5\right)^{2}} + \frac{4 x^{3} + 3}{3 x^{4} + 5} = 0$$
Solve this equationThe roots of this equation
$$x_{1} = \frac{\sqrt{- \frac{10}{27 \sqrt[3]{- \frac{20}{6561} + \frac{5 \sqrt{10951}}{6561}}} + \frac{16}{729} + 2 \sqrt[3]{- \frac{20}{6561} + \frac{5 \sqrt{10951}}{6561}}}}{2} + \frac{2}{27} + \frac{\sqrt{- 2 \sqrt[3]{- \frac{20}{6561} + \frac{5 \sqrt{10951}}{6561}} + \frac{32}{729} + \frac{10}{27 \sqrt[3]{- \frac{20}{6561} + \frac{5 \sqrt{10951}}{6561}}} + \frac{128}{19683 \sqrt{- \frac{10}{27 \sqrt[3]{- \frac{20}{6561} + \frac{5 \sqrt{10951}}{6561}}} + \frac{16}{729} + 2 \sqrt[3]{- \frac{20}{6561} + \frac{5 \sqrt{10951}}{6561}}}}}}{2}$$
$$x_{2} = - \frac{\sqrt{- 2 \sqrt[3]{- \frac{20}{6561} + \frac{5 \sqrt{10951}}{6561}} + \frac{32}{729} + \frac{10}{27 \sqrt[3]{- \frac{20}{6561} + \frac{5 \sqrt{10951}}{6561}}} + \frac{128}{19683 \sqrt{- \frac{10}{27 \sqrt[3]{- \frac{20}{6561} + \frac{5 \sqrt{10951}}{6561}}} + \frac{16}{729} + 2 \sqrt[3]{- \frac{20}{6561} + \frac{5 \sqrt{10951}}{6561}}}}}}{2} + \frac{\sqrt{- \frac{10}{27 \sqrt[3]{- \frac{20}{6561} + \frac{5 \sqrt{10951}}{6561}}} + \frac{16}{729} + 2 \sqrt[3]{- \frac{20}{6561} + \frac{5 \sqrt{10951}}{6561}}}}{2} + \frac{2}{27}$$
The values of the extrema at the points:
4
/ _______________________________________________________________________________________________________________________________________________________________\ _______________________________________________________________________________________________________________________________________________________________
| / ______________________ | / ______________________
_______________________________________________________________________________________________________________________________________________________________ | / / _______ | / / _______
/ ______________________ | / 32 / 20 5*\/ 10951 10 128 | / 32 / 20 5*\/ 10951 10 128
/ / _______ | / --- - 2*3 / - ---- + ----------- + ------------------------------ + -------------------------------------------------------------------------------------- | 3* / --- - 2*3 / - ---- + ----------- + ------------------------------ + --------------------------------------------------------------------------------------
/ 32 / 20 5*\/ 10951 10 128 | ______________________________________________________________________ / 729 \/ 6561 6561 ______________________ ______________________________________________________________________ | ______________________________________________________________________ / 729 \/ 6561 6561 ______________________ ______________________________________________________________________
/ --- - 2*3 / - ---- + ----------- + ------------------------------ + -------------------------------------------------------------------------------------- | / ______________________ / / _______ / ______________________ | / ______________________ / / _______ / ______________________
______________________________________________________________________ / 729 \/ 6561 6561 ______________________ ______________________________________________________________________ | / / _______ / / 20 5*\/ 10951 / / _______ | / / _______ / / 20 5*\/ 10951 / / _______
/ ______________________ / / _______ / ______________________ | / 16 / 20 5*\/ 10951 10 / 27*3 / - ---- + ----------- / 16 / 20 5*\/ 10951 10 | / 16 / 20 5*\/ 10951 10 / 27*3 / - ---- + ----------- / 16 / 20 5*\/ 10951 10
/ / _______ / / 20 5*\/ 10951 / / _______ | / --- + 2*3 / - ---- + ----------- - ------------------------------ / \/ 6561 6561 19683* / --- + 2*3 / - ---- + ----------- - ------------------------------ | 3* / --- + 2*3 / - ---- + ----------- - ------------------------------ / \/ 6561 6561 19683* / --- + 2*3 / - ---- + ----------- - ------------------------------
/ 16 / 20 5*\/ 10951 10 / 27*3 / - ---- + ----------- / 16 / 20 5*\/ 10951 10 | / 729 \/ 6561 6561 ______________________ / / 729 \/ 6561 6561 ______________________ | / 729 \/ 6561 6561 ______________________ / / 729 \/ 6561 6561 ______________________
/ --- + 2*3 / - ---- + ----------- - ------------------------------ / \/ 6561 6561 19683* / --- + 2*3 / - ---- + ----------- - ------------------------------ | / / _______ / / / _______ | / / _______ / / / _______
/ 729 \/ 6561 6561 ______________________ / / 729 \/ 6561 6561 ______________________ | / / 20 5*\/ 10951 / / / 20 5*\/ 10951 | / / 20 5*\/ 10951 / / / 20 5*\/ 10951
/ / _______ / / / _______ | / 27*3 / - ---- + ----------- / / 27*3 / - ---- + ----------- | / 27*3 / - ---- + ----------- / / 27*3 / - ---- + -----------
/ / 20 5*\/ 10951 / / / 20 5*\/ 10951 11 |2 \/ \/ 6561 6561 \/ \/ \/ 6561 6561 | \/ \/ 6561 6561 \/ \/ \/ 6561 6561
/ 27*3 / - ---- + ----------- / / 27*3 / - ---- + ----------- -- + |-- + -------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| + ---------------------------------------------------------------------------------- + --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
2 \/ \/ 6561 6561 \/ \/ \/ 6561 6561 9 \27 2 2 / 2 2
(-- + -------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------, ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------)
27 2 2 4
/ _______________________________________________________________________________________________________________________________________________________________\
| / ______________________ |
| / / _______ |
| / 32 / 20 5*\/ 10951 10 128 |
| / --- - 2*3 / - ---- + ----------- + ------------------------------ + -------------------------------------------------------------------------------------- |
| ______________________________________________________________________ / 729 \/ 6561 6561 ______________________ ______________________________________________________________________ |
| / ______________________ / / _______ / ______________________ |
| / / _______ / / 20 5*\/ 10951 / / _______ |
| / 16 / 20 5*\/ 10951 10 / 27*3 / - ---- + ----------- / 16 / 20 5*\/ 10951 10 |
| / --- + 2*3 / - ---- + ----------- - ------------------------------ / \/ 6561 6561 19683* / --- + 2*3 / - ---- + ----------- - ------------------------------ |
| / 729 \/ 6561 6561 ______________________ / / 729 \/ 6561 6561 ______________________ |
| / / _______ / / / _______ |
| / / 20 5*\/ 10951 / / / 20 5*\/ 10951 |
| / 27*3 / - ---- + ----------- / / 27*3 / - ---- + ----------- |
|2 \/ \/ 6561 6561 \/ \/ \/ 6561 6561 |
5 + 3*|-- + -------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|
\27 2 2 /
4
/ _______________________________________________________________________________________________________________________________________________________________\ _______________________________________________________________________________________________________________________________________________________________
| / ______________________ | / ______________________
_______________________________________________________________________________________________________________________________________________________________ | / / _______ | / / _______
/ ______________________ | / 32 / 20 5*\/ 10951 10 128 | / 32 / 20 5*\/ 10951 10 128
/ / _______ | / --- - 2*3 / - ---- + ----------- + ------------------------------ + -------------------------------------------------------------------------------------- | 3* / --- - 2*3 / - ---- + ----------- + ------------------------------ + --------------------------------------------------------------------------------------
/ 32 / 20 5*\/ 10951 10 128 | ______________________________________________________________________ / 729 \/ 6561 6561 ______________________ ______________________________________________________________________ | / 729 \/ 6561 6561 ______________________ ______________________________________________________________________ ______________________________________________________________________
/ --- - 2*3 / - ---- + ----------- + ------------------------------ + -------------------------------------------------------------------------------------- | / ______________________ / / _______ / ______________________ | / / _______ / ______________________ / ______________________
______________________________________________________________________ / 729 \/ 6561 6561 ______________________ ______________________________________________________________________ | / / _______ / / 20 5*\/ 10951 / / _______ | / / 20 5*\/ 10951 / / _______ / / _______
/ ______________________ / / _______ / ______________________ | / 16 / 20 5*\/ 10951 10 / 27*3 / - ---- + ----------- / 16 / 20 5*\/ 10951 10 | / 27*3 / - ---- + ----------- / 16 / 20 5*\/ 10951 10 / 16 / 20 5*\/ 10951 10
/ / _______ / / 20 5*\/ 10951 / / _______ | / --- + 2*3 / - ---- + ----------- - ------------------------------ / \/ 6561 6561 19683* / --- + 2*3 / - ---- + ----------- - ------------------------------ | / \/ 6561 6561 19683* / --- + 2*3 / - ---- + ----------- - ------------------------------ 3* / --- + 2*3 / - ---- + ----------- - ------------------------------
/ 16 / 20 5*\/ 10951 10 / 27*3 / - ---- + ----------- / 16 / 20 5*\/ 10951 10 | / 729 \/ 6561 6561 ______________________ / / 729 \/ 6561 6561 ______________________ | / / 729 \/ 6561 6561 ______________________ / 729 \/ 6561 6561 ______________________
/ --- + 2*3 / - ---- + ----------- - ------------------------------ / \/ 6561 6561 19683* / --- + 2*3 / - ---- + ----------- - ------------------------------ | / / _______ / / / _______ | / / / _______ / / _______
/ 729 \/ 6561 6561 ______________________ / / 729 \/ 6561 6561 ______________________ | / / 20 5*\/ 10951 / / / 20 5*\/ 10951 | / / / 20 5*\/ 10951 / / 20 5*\/ 10951
/ / _______ / / / _______ | / 27*3 / - ---- + ----------- / / 27*3 / - ---- + ----------- | / / 27*3 / - ---- + ----------- / 27*3 / - ---- + -----------
/ / 20 5*\/ 10951 / / / 20 5*\/ 10951 11 |2 \/ \/ 6561 6561 \/ \/ \/ 6561 6561 | \/ \/ \/ 6561 6561 \/ \/ 6561 6561
/ 27*3 / - ---- + ----------- / / 27*3 / - ---- + ----------- -- + |-- + -------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| - -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ----------------------------------------------------------------------------------
2 \/ \/ 6561 6561 \/ \/ \/ 6561 6561 9 \27 2 2 / 2 2
(-- + -------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------, ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------)
27 2 2 4
/ _______________________________________________________________________________________________________________________________________________________________\
| / ______________________ |
| / / _______ |
| / 32 / 20 5*\/ 10951 10 128 |
| / --- - 2*3 / - ---- + ----------- + ------------------------------ + -------------------------------------------------------------------------------------- |
| ______________________________________________________________________ / 729 \/ 6561 6561 ______________________ ______________________________________________________________________ |
| / ______________________ / / _______ / ______________________ |
| / / _______ / / 20 5*\/ 10951 / / _______ |
| / 16 / 20 5*\/ 10951 10 / 27*3 / - ---- + ----------- / 16 / 20 5*\/ 10951 10 |
| / --- + 2*3 / - ---- + ----------- - ------------------------------ / \/ 6561 6561 19683* / --- + 2*3 / - ---- + ----------- - ------------------------------ |
| / 729 \/ 6561 6561 ______________________ / / 729 \/ 6561 6561 ______________________ |
| / / _______ / / / _______ |
| / / 20 5*\/ 10951 / / / 20 5*\/ 10951 |
| / 27*3 / - ---- + ----------- / / 27*3 / - ---- + ----------- |
|2 \/ \/ 6561 6561 \/ \/ \/ 6561 6561 |
5 + 3*|-- + -------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|
\27 2 2 /
Intervals of increase and decrease of the function:Let's find intervals where the function increases and decreases, as well as minima and maxima of the function, for this let's look how the function behaves itself in the extremas and at the slightest deviation from:
Minima of the function at points:
$$x_{1} = - \frac{\sqrt{\left|{- 2 \sqrt[3]{- \frac{20}{6561} + \frac{5 \sqrt{10951}}{6561}} + \frac{32}{729} + \frac{10}{27 \sqrt[3]{- \frac{20}{6561} + \frac{5 \sqrt{10951}}{6561}}} + \frac{128}{19683 \sqrt{- \frac{10}{27 \sqrt[3]{- \frac{20}{6561} + \frac{5 \sqrt{10951}}{6561}}} + \frac{16}{729} + 2 \sqrt[3]{- \frac{20}{6561} + \frac{5 \sqrt{10951}}{6561}}}}}\right|}}{2} + \frac{\sqrt{- \frac{10}{27 \sqrt[3]{- \frac{20}{6561} + \frac{5 \sqrt{10951}}{6561}}} + \frac{16}{729} + 2 \sqrt[3]{- \frac{20}{6561} + \frac{5 \sqrt{10951}}{6561}}}}{2} + \frac{2}{27}$$
Maxima of the function at points:
$$x_{1} = \frac{\sqrt{- \frac{10}{27 \sqrt[3]{- \frac{20}{6561} + \frac{5 \sqrt{10951}}{6561}}} + \frac{16}{729} + 2 \sqrt[3]{- \frac{20}{6561} + \frac{5 \sqrt{10951}}{6561}}}}{2} + \frac{2}{27} + \frac{\sqrt{\left|{- 2 \sqrt[3]{- \frac{20}{6561} + \frac{5 \sqrt{10951}}{6561}} + \frac{32}{729} + \frac{10}{27 \sqrt[3]{- \frac{20}{6561} + \frac{5 \sqrt{10951}}{6561}}} + \frac{128}{19683 \sqrt{- \frac{10}{27 \sqrt[3]{- \frac{20}{6561} + \frac{5 \sqrt{10951}}{6561}}} + \frac{16}{729} + 2 \sqrt[3]{- \frac{20}{6561} + \frac{5 \sqrt{10951}}{6561}}}}}\right|}}{2}$$
Decreasing at intervals
$$\left[- \frac{\sqrt{\left|{- 2 \sqrt[3]{- \frac{20}{6561} + \frac{5 \sqrt{10951}}{6561}} + \frac{32}{729} + \frac{10}{27 \sqrt[3]{- \frac{20}{6561} + \frac{5 \sqrt{10951}}{6561}}} + \frac{128}{19683 \sqrt{- \frac{10}{27 \sqrt[3]{- \frac{20}{6561} + \frac{5 \sqrt{10951}}{6561}}} + \frac{16}{729} + 2 \sqrt[3]{- \frac{20}{6561} + \frac{5 \sqrt{10951}}{6561}}}}}\right|}}{2} + \frac{\sqrt{- \frac{10}{27 \sqrt[3]{- \frac{20}{6561} + \frac{5 \sqrt{10951}}{6561}}} + \frac{16}{729} + 2 \sqrt[3]{- \frac{20}{6561} + \frac{5 \sqrt{10951}}{6561}}}}{2} + \frac{2}{27}, \frac{\sqrt{- \frac{10}{27 \sqrt[3]{- \frac{20}{6561} + \frac{5 \sqrt{10951}}{6561}}} + \frac{16}{729} + 2 \sqrt[3]{- \frac{20}{6561} + \frac{5 \sqrt{10951}}{6561}}}}{2} + \frac{2}{27} + \frac{\sqrt{\left|{- 2 \sqrt[3]{- \frac{20}{6561} + \frac{5 \sqrt{10951}}{6561}} + \frac{32}{729} + \frac{10}{27 \sqrt[3]{- \frac{20}{6561} + \frac{5 \sqrt{10951}}{6561}}} + \frac{128}{19683 \sqrt{- \frac{10}{27 \sqrt[3]{- \frac{20}{6561} + \frac{5 \sqrt{10951}}{6561}}} + \frac{16}{729} + 2 \sqrt[3]{- \frac{20}{6561} + \frac{5 \sqrt{10951}}{6561}}}}}\right|}}{2}\right]$$
Increasing at intervals
$$\left(-\infty, - \frac{\sqrt{\left|{- 2 \sqrt[3]{- \frac{20}{6561} + \frac{5 \sqrt{10951}}{6561}} + \frac{32}{729} + \frac{10}{27 \sqrt[3]{- \frac{20}{6561} + \frac{5 \sqrt{10951}}{6561}}} + \frac{128}{19683 \sqrt{- \frac{10}{27 \sqrt[3]{- \frac{20}{6561} + \frac{5 \sqrt{10951}}{6561}}} + \frac{16}{729} + 2 \sqrt[3]{- \frac{20}{6561} + \frac{5 \sqrt{10951}}{6561}}}}}\right|}}{2} + \frac{\sqrt{- \frac{10}{27 \sqrt[3]{- \frac{20}{6561} + \frac{5 \sqrt{10951}}{6561}}} + \frac{16}{729} + 2 \sqrt[3]{- \frac{20}{6561} + \frac{5 \sqrt{10951}}{6561}}}}{2} + \frac{2}{27}\right] \cup \left[\frac{\sqrt{- \frac{10}{27 \sqrt[3]{- \frac{20}{6561} + \frac{5 \sqrt{10951}}{6561}}} + \frac{16}{729} + 2 \sqrt[3]{- \frac{20}{6561} + \frac{5 \sqrt{10951}}{6561}}}}{2} + \frac{2}{27} + \frac{\sqrt{\left|{- 2 \sqrt[3]{- \frac{20}{6561} + \frac{5 \sqrt{10951}}{6561}} + \frac{32}{729} + \frac{10}{27 \sqrt[3]{- \frac{20}{6561} + \frac{5 \sqrt{10951}}{6561}}} + \frac{128}{19683 \sqrt{- \frac{10}{27 \sqrt[3]{- \frac{20}{6561} + \frac{5 \sqrt{10951}}{6561}}} + \frac{16}{729} + 2 \sqrt[3]{- \frac{20}{6561} + \frac{5 \sqrt{10951}}{6561}}}}}\right|}}{2}, \infty\right)$$