Mister Exam
Graphing y = ctg(sqrt(x+lncosx))
The solution
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/ _________________\ f(x) = cot\\/ x + log(cos(x)) /
$$f{\left(x \right)} = \cot{\left(\sqrt{x + \log{\left(\cos{\left(x \right)} \right)}} \right)}$$
f = cot(sqrt(x + log(cos(x))))
The graph of the function
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
$$\frac{\left(- \frac{\sin{\left(x \right)}}{2 \cos{\left(x \right)}} + \frac{1}{2}\right) \left(- \cot^{2}{\left(\sqrt{x + \log{\left(\cos{\left(x \right)} \right)}} \right)} - 1\right)}{\sqrt{x + \log{\left(\cos{\left(x \right)} \right)}}} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = - \frac{3 \pi}{4}$$
$$x_{2} = \frac{\pi}{4}$$
The values of the extrema at the points:
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:
$$x_{1} = \frac{\pi}{4}$$
Maxima of the function at points:
$$x_{1} = - \frac{3 \pi}{4}$$
Decreasing at intervals
$$\left(-\infty, - \frac{3 \pi}{4}\right] \cup \left[\frac{\pi}{4}, \infty\right)$$
Increasing at intervals
$$\left[- \frac{3 \pi}{4}, \frac{\pi}{4}\right]$$
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(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
$$\frac{\left(- \frac{\sin{\left(x \right)}}{2 \cos{\left(x \right)}} + \frac{1}{2}\right) \left(- \cot^{2}{\left(\sqrt{x + \log{\left(\cos{\left(x \right)} \right)}} \right)} - 1\right)}{\sqrt{x + \log{\left(\cos{\left(x \right)} \right)}}} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = - \frac{3 \pi}{4}$$
$$x_{2} = \frac{\pi}{4}$$
The values of the extrema at the points:
/ ____________________________\
| / / ___\ |
-3*pi | / 3*pi |\/ 2 | |
(-----, cot| / - ---- + pi*I + log|-----| |)
4 \\/ 4 \ 2 / / / _________________\
| / / ___\ |
pi | / pi |\/ 2 | |
(--, cot| / -- + log|-----| |)
4 \\/ 4 \ 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:
$$x_{1} = \frac{\pi}{4}$$
Maxima of the function at points:
$$x_{1} = - \frac{3 \pi}{4}$$
Decreasing at intervals
$$\left(-\infty, - \frac{3 \pi}{4}\right] \cup \left[\frac{\pi}{4}, \infty\right)$$
Increasing at intervals
$$\left[- \frac{3 \pi}{4}, \frac{\pi}{4}\right]$$
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:
$$\cot{\left(\sqrt{x + \log{\left(\cos{\left(x \right)} \right)}} \right)} = \cot{\left(\sqrt{- x + \log{\left(\cos{\left(x \right)} \right)}} \right)}$$
- No
$$\cot{\left(\sqrt{x + \log{\left(\cos{\left(x \right)} \right)}} \right)} = - \cot{\left(\sqrt{- x + \log{\left(\cos{\left(x \right)} \right)}} \right)}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\cot{\left(\sqrt{x + \log{\left(\cos{\left(x \right)} \right)}} \right)} = \cot{\left(\sqrt{- x + \log{\left(\cos{\left(x \right)} \right)}} \right)}$$
- No
$$\cot{\left(\sqrt{x + \log{\left(\cos{\left(x \right)} \right)}} \right)} = - \cot{\left(\sqrt{- x + \log{\left(\cos{\left(x \right)} \right)}} \right)}$$
- No
so, the function
not is
neither even, nor odd