Question

$f\left(x\right)=\tan x-{\tan }^{2}x+{\tan }^{3}x,x\ne n\pi +\frac{\pi }{2}$

• A. Is monotonically increasing
• B. Is monotonically decreasing
• C. has a point of maxima
• D. has a point of minimum

Answer 1

The correct option is A Is monotonically increasing

The given equation is:

$f\left(x\right)=\tan x-{\tan }^{2}x+{\tan }^{3}x$

Differentiating once to find the slope of the function.

$\Rightarrow f^{\prime }\left(x\right)={\sec }^{2}x-2\tan x{\sec }^{2}x+3{\tan }^{2}x{\sec }^{2}x$

$\Rightarrow f^{\prime }\left(x\right)={\sec }^{2}x(1-2\tan x+3{\tan }^{2}x)$

${\sec }^{2}x>0$. This is always true as it a square functiion.

$(1-2\tan x+3{\tan }^{2}x)>0$

The quadratic roots of teh above equation gives the following result. $D=\sqrt{b^2-4ac}=\sqrt{4-12}$. Tt has complex roots and no real roots. Thus the function is always greater than 0.

Thus $f^{\prime }\left(x\right)>0$

Hence the function is increasing monotonically. ....Answer