Mister Exam
Graphing y = (ln((4x-3)/x))/x
The solution
You have entered
[src]
/4*x - 3\
log|-------|
\ x /
f(x) = ------------
x
$$f{\left(x \right)} = \frac{\log{\left(\frac{4 x - 3}{x} \right)}}{x}$$
f = log((4*x - 3)/x)/x
The graph of the function
The domain of the function
The points at which the function is not precisely defined:
$$x_{1} = 0$$
$$x_{1} = 0$$
The points of intersection with the X-axis coordinate
Graph of the function intersects the axis X at f = 0
so we need to solve the equation:
$$\frac{\log{\left(\frac{4 x - 3}{x} \right)}}{x} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 1$$
Numerical solution
$$x_{1} = 1$$
so we need to solve the equation:
$$\frac{\log{\left(\frac{4 x - 3}{x} \right)}}{x} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 1$$
Numerical solution
$$x_{1} = 1$$
Extrema of the function
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
$$\frac{\frac{4}{x} - \frac{4 x - 3}{x^{2}}}{4 x - 3} - \frac{\log{\left(\frac{4 x - 3}{x} \right)}}{x^{2}} = 0$$
Solve this equation
Solutions are not found,
function may have no extrema
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(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
$$\frac{\frac{4}{x} - \frac{4 x - 3}{x^{2}}}{4 x - 3} - \frac{\log{\left(\frac{4 x - 3}{x} \right)}}{x^{2}} = 0$$
Solve this equation
Solutions are not found,
function may have no extrema
Inflection points
Let's find the inflection points, we'll need to solve the equation for this
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = $$
the second derivative
$$\frac{- \frac{\left(4 - \frac{4 x - 3}{x}\right) \left(\frac{4}{4 x - 3} + \frac{1}{x}\right)}{4 x - 3} - \frac{2 \left(4 - \frac{4 x - 3}{x}\right)}{x \left(4 x - 3\right)} + \frac{2 \log{\left(\frac{4 x - 3}{x} \right)}}{x^{2}}}{x} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = -34463.1617445174$$
$$x_{2} = 29120.2375861495$$
$$x_{3} = -31073.3486407714$$
$$x_{4} = -31920.7903204858$$
$$x_{5} = 32509.8599370855$$
$$x_{6} = 30815.0260132865$$
$$x_{7} = 21494.5586034282$$
$$x_{8} = 35899.6375951767$$
$$x_{9} = -14128.2973280725$$
$$x_{10} = -37853.0797396681$$
$$x_{11} = 22341.7578786277$$
$$x_{12} = -40395.5716347875$$
$$x_{13} = -20905.0218512504$$
$$x_{14} = 13872.2073416956$$
$$x_{15} = 15565.5036315266$$
$$x_{16} = -30225.9157351273$$
$$x_{17} = -25988.9115623932$$
$$x_{18} = 28272.8631057757$$
$$x_{19} = -28531.0792499336$$
$$x_{20} = 35052.1811732323$$
$$x_{21} = 18106.1679839655$$
$$x_{22} = 16412.311833728$$
$$x_{23} = -22599.5692529926$$
$$x_{24} = -35310.6325517887$$
$$x_{25} = 33357.2917110184$$
$$x_{26} = -19210.5843984905$$
$$x_{27} = 27425.5034823996$$
$$x_{28} = 42679.5064716985$$
$$x_{29} = 40137.0162497684$$
$$x_{30} = 40984.5084095103$$
$$x_{31} = -13281.4779669118$$
$$x_{32} = 17259.2041757847$$
$$x_{33} = -36158.1094164695$$
$$x_{34} = -37005.5919352029$$
$$x_{35} = -14975.199112923$$
$$x_{36} = -25141.5500291495$$
$$x_{37} = -18363.4158484489$$
$$x_{38} = 39289.5290787557$$
$$x_{39} = -21752.2833329525$$
$$x_{40} = -38700.5724928573$$
$$x_{41} = 13025.765269842$$
$$x_{42} = 14718.7954291455$$
$$x_{43} = 36747.1010674695$$
$$x_{44} = -23446.8770873619$$
$$x_{45} = -24294.204648097$$
$$x_{46} = -33615.6974373164$$
$$x_{47} = -39548.0698858168$$
$$x_{48} = 26578.1602054129$$
$$x_{49} = 2.40091989837846$$
$$x_{50} = 24883.5297149924$$
$$x_{51} = -27683.6773928884$$
$$x_{52} = 19800.2708714673$$
$$x_{53} = -15822.1709824582$$
$$x_{54} = -32768.2401170578$$
$$x_{55} = 20647.394656132$$
$$x_{56} = -41243.0774786877$$
$$x_{57} = 31662.4377629901$$
$$x_{58} = -26836.2877816212$$
$$x_{59} = -17516.2868189532$$
$$x_{60} = 34204.7323449675$$
$$x_{61} = 37594.5710974087$$
$$x_{62} = 18953.193008733$$
$$x_{63} = 23188.9883931729$$
$$x_{64} = 38442.0472371585$$
$$x_{65} = -42090.5871768921$$
$$x_{66} = -20057.7877350434$$
$$x_{67} = -29378.4923336919$$
$$x_{68} = -16669.2029440586$$
$$x_{69} = 41832.0052457287$$
$$x_{70} = 25730.8349701928$$
$$x_{71} = 29967.6256097513$$
$$x_{72} = 24036.246666055$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = 0$$
$$\lim_{x \to 0^-}\left(\frac{- \frac{\left(4 - \frac{4 x - 3}{x}\right) \left(\frac{4}{4 x - 3} + \frac{1}{x}\right)}{4 x - 3} - \frac{2 \left(4 - \frac{4 x - 3}{x}\right)}{x \left(4 x - 3\right)} + \frac{2 \log{\left(\frac{4 x - 3}{x} \right)}}{x^{2}}}{x}\right) = -\infty$$
$$\lim_{x \to 0^+}\left(\frac{- \frac{\left(4 - \frac{4 x - 3}{x}\right) \left(\frac{4}{4 x - 3} + \frac{1}{x}\right)}{4 x - 3} - \frac{2 \left(4 - \frac{4 x - 3}{x}\right)}{x \left(4 x - 3\right)} + \frac{2 \log{\left(\frac{4 x - 3}{x} \right)}}{x^{2}}}{x}\right) = \infty$$
- the limits are not equal, so
$$x_{1} = 0$$
- is an inflection point
Сonvexity and concavity intervals:
Let’s find the intervals where the function is convex or concave, for this look at the behaviour of the function at the inflection points:
Concave at the intervals
$$\left[2.40091989837846, \infty\right)$$
Convex at the intervals
$$\left(-\infty, 2.40091989837846\right]$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = $$
the second derivative
$$\frac{- \frac{\left(4 - \frac{4 x - 3}{x}\right) \left(\frac{4}{4 x - 3} + \frac{1}{x}\right)}{4 x - 3} - \frac{2 \left(4 - \frac{4 x - 3}{x}\right)}{x \left(4 x - 3\right)} + \frac{2 \log{\left(\frac{4 x - 3}{x} \right)}}{x^{2}}}{x} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = -34463.1617445174$$
$$x_{2} = 29120.2375861495$$
$$x_{3} = -31073.3486407714$$
$$x_{4} = -31920.7903204858$$
$$x_{5} = 32509.8599370855$$
$$x_{6} = 30815.0260132865$$
$$x_{7} = 21494.5586034282$$
$$x_{8} = 35899.6375951767$$
$$x_{9} = -14128.2973280725$$
$$x_{10} = -37853.0797396681$$
$$x_{11} = 22341.7578786277$$
$$x_{12} = -40395.5716347875$$
$$x_{13} = -20905.0218512504$$
$$x_{14} = 13872.2073416956$$
$$x_{15} = 15565.5036315266$$
$$x_{16} = -30225.9157351273$$
$$x_{17} = -25988.9115623932$$
$$x_{18} = 28272.8631057757$$
$$x_{19} = -28531.0792499336$$
$$x_{20} = 35052.1811732323$$
$$x_{21} = 18106.1679839655$$
$$x_{22} = 16412.311833728$$
$$x_{23} = -22599.5692529926$$
$$x_{24} = -35310.6325517887$$
$$x_{25} = 33357.2917110184$$
$$x_{26} = -19210.5843984905$$
$$x_{27} = 27425.5034823996$$
$$x_{28} = 42679.5064716985$$
$$x_{29} = 40137.0162497684$$
$$x_{30} = 40984.5084095103$$
$$x_{31} = -13281.4779669118$$
$$x_{32} = 17259.2041757847$$
$$x_{33} = -36158.1094164695$$
$$x_{34} = -37005.5919352029$$
$$x_{35} = -14975.199112923$$
$$x_{36} = -25141.5500291495$$
$$x_{37} = -18363.4158484489$$
$$x_{38} = 39289.5290787557$$
$$x_{39} = -21752.2833329525$$
$$x_{40} = -38700.5724928573$$
$$x_{41} = 13025.765269842$$
$$x_{42} = 14718.7954291455$$
$$x_{43} = 36747.1010674695$$
$$x_{44} = -23446.8770873619$$
$$x_{45} = -24294.204648097$$
$$x_{46} = -33615.6974373164$$
$$x_{47} = -39548.0698858168$$
$$x_{48} = 26578.1602054129$$
$$x_{49} = 2.40091989837846$$
$$x_{50} = 24883.5297149924$$
$$x_{51} = -27683.6773928884$$
$$x_{52} = 19800.2708714673$$
$$x_{53} = -15822.1709824582$$
$$x_{54} = -32768.2401170578$$
$$x_{55} = 20647.394656132$$
$$x_{56} = -41243.0774786877$$
$$x_{57} = 31662.4377629901$$
$$x_{58} = -26836.2877816212$$
$$x_{59} = -17516.2868189532$$
$$x_{60} = 34204.7323449675$$
$$x_{61} = 37594.5710974087$$
$$x_{62} = 18953.193008733$$
$$x_{63} = 23188.9883931729$$
$$x_{64} = 38442.0472371585$$
$$x_{65} = -42090.5871768921$$
$$x_{66} = -20057.7877350434$$
$$x_{67} = -29378.4923336919$$
$$x_{68} = -16669.2029440586$$
$$x_{69} = 41832.0052457287$$
$$x_{70} = 25730.8349701928$$
$$x_{71} = 29967.6256097513$$
$$x_{72} = 24036.246666055$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = 0$$
$$\lim_{x \to 0^-}\left(\frac{- \frac{\left(4 - \frac{4 x - 3}{x}\right) \left(\frac{4}{4 x - 3} + \frac{1}{x}\right)}{4 x - 3} - \frac{2 \left(4 - \frac{4 x - 3}{x}\right)}{x \left(4 x - 3\right)} + \frac{2 \log{\left(\frac{4 x - 3}{x} \right)}}{x^{2}}}{x}\right) = -\infty$$
$$\lim_{x \to 0^+}\left(\frac{- \frac{\left(4 - \frac{4 x - 3}{x}\right) \left(\frac{4}{4 x - 3} + \frac{1}{x}\right)}{4 x - 3} - \frac{2 \left(4 - \frac{4 x - 3}{x}\right)}{x \left(4 x - 3\right)} + \frac{2 \log{\left(\frac{4 x - 3}{x} \right)}}{x^{2}}}{x}\right) = \infty$$
- the limits are not equal, so
$$x_{1} = 0$$
- is an inflection point
Сonvexity and concavity intervals:
Let’s find the intervals where the function is convex or concave, for this look at the behaviour of the function at the inflection points:
Concave at the intervals
$$\left[2.40091989837846, \infty\right)$$
Convex at the intervals
$$\left(-\infty, 2.40091989837846\right]$$
Vertical asymptotes
Have:
$$x_{1} = 0$$
$$x_{1} = 0$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(\frac{\log{\left(\frac{4 x - 3}{x} \right)}}{x}\right) = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 0$$
$$\lim_{x \to \infty}\left(\frac{\log{\left(\frac{4 x - 3}{x} \right)}}{x}\right) = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = 0$$
$$\lim_{x \to -\infty}\left(\frac{\log{\left(\frac{4 x - 3}{x} \right)}}{x}\right) = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 0$$
$$\lim_{x \to \infty}\left(\frac{\log{\left(\frac{4 x - 3}{x} \right)}}{x}\right) = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = 0$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of log((4*x - 3)/x)/x, divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\log{\left(\frac{4 x - 3}{x} \right)}}{x^{2}}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\log{\left(\frac{4 x - 3}{x} \right)}}{x^{2}}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{\log{\left(\frac{4 x - 3}{x} \right)}}{x^{2}}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\log{\left(\frac{4 x - 3}{x} \right)}}{x^{2}}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
Even and odd functions
Let's check, whether the function even or odd by using relations f = f(-x) и f = -f(-x).
So, check:
$$\frac{\log{\left(\frac{4 x - 3}{x} \right)}}{x} = - \frac{\log{\left(- \frac{- 4 x - 3}{x} \right)}}{x}$$
- No
$$\frac{\log{\left(\frac{4 x - 3}{x} \right)}}{x} = \frac{\log{\left(- \frac{- 4 x - 3}{x} \right)}}{x}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\frac{\log{\left(\frac{4 x - 3}{x} \right)}}{x} = - \frac{\log{\left(- \frac{- 4 x - 3}{x} \right)}}{x}$$
- No
$$\frac{\log{\left(\frac{4 x - 3}{x} \right)}}{x} = \frac{\log{\left(- \frac{- 4 x - 3}{x} \right)}}{x}$$
- No
so, the function
not is
neither even, nor odd