Mister Exam

# Graphing y = (sqrt(cos(x))*cos(75x)+sqrt(abs(x))-0,7)*(4-x*x)^0.2,sqrt(9-x^2),-sqrt(9-x^2),-x

Function f()

from to

does show?

### The solution

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                                                              ________      ________
/  ________               _____   7 \ 5 _________    /      2      /      2
f(x) = (|\/ cos(x) *cos(75*x) + \/ |x|  - --|*\/ 4 - x*x , \/  9 - x  , -\/  9 - x  , -x)
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$$f{\left(x \right)} = \left( \sqrt[5]{- x x + 4} \left(\left(\sqrt{\cos{\left(x \right)}} \cos{\left(75 x \right)} + \sqrt{\left|{x}\right|}\right) - \frac{7}{10}\right), \ \sqrt{9 - x^{2}}, \ - \sqrt{9 - x^{2}}, \ - x\right)$$
f = ((-x*x + 4)^(1/5)*(sqrt(cos(x))*cos(75*x) + sqrt(|x|) - 7/10, sqrt(9 - x^2), -sqrt(9 - x^2), -x))
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:
$$\left( \sqrt[5]{- x x + 4} \left(\left(\sqrt{\cos{\left(x \right)}} \cos{\left(75 x \right)} + \sqrt{\left|{x}\right|}\right) - \frac{7}{10}\right), \ \sqrt{9 - x^{2}}, \ - \sqrt{9 - x^{2}}, \ - x\right) = 0$$
Solve this equation
it's possible that the graph doesn't intersect the axis X
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to ((-x*x + 4)^(1/5)*(sqrt(cos(x))*cos(75*x) + sqrt(|x|) - 7/10), sqrt(9 - x^2), -sqrt(9 - x^2), -x).
                                                       ________      ________
/  ________               _____   7 \ 5 _________    /      2      /      2
(|\/ cos(0) *cos(75*0) + \/ |0|  - --|*\/ 4 - 0*0, \/  9 - 0 , -\/  9 - 0 , -0)
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The result:
$$f{\left(0 \right)} = \left( \frac{3 \cdot 2^{\frac{2}{5}}}{10}, \ 3, \ -3, \ 0\right)$$
The point:
(0, (3*2^(2/5)/10, 3, -3, 0))
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
$$\frac{d}{d x} \left( \sqrt[5]{- x x + 4} \left(\left(\sqrt{\cos{\left(x \right)}} \cos{\left(75 x \right)} + \sqrt{\left|{x}\right|}\right) - \frac{7}{10}\right), \ \sqrt{9 - x^{2}}, \ - \sqrt{9 - x^{2}}, \ - x\right) = 0$$
Solve this equation
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
$$\frac{d^{2}}{d x^{2}} \left( \sqrt[5]{- x x + 4} \left(\left(\sqrt{\cos{\left(x \right)}} \cos{\left(75 x \right)} + \sqrt{\left|{x}\right|}\right) - \frac{7}{10}\right), \ \sqrt{9 - x^{2}}, \ - \sqrt{9 - x^{2}}, \ - x\right) = 0$$
Solve this equation
$$\lim_{x \to -\infty} \left( \sqrt[5]{- x x + 4} \left(\left(\sqrt{\cos{\left(x \right)}} \cos{\left(75 x \right)} + \sqrt{\left|{x}\right|}\right) - \frac{7}{10}\right), \ \sqrt{9 - x^{2}}, \ - \sqrt{9 - x^{2}}, \ - x\right)$$
$$\lim_{x \to \infty} \left( \sqrt[5]{- x x + 4} \left(\left(\sqrt{\cos{\left(x \right)}} \cos{\left(75 x \right)} + \sqrt{\left|{x}\right|}\right) - \frac{7}{10}\right), \ \sqrt{9 - x^{2}}, \ - \sqrt{9 - x^{2}}, \ - x\right)$$