Question

$f\left(x\right)=4\tan x-{\tan }^{2}x+{\tan }^{3}x,x\ne n\pi +\frac{\pi }{2}$

• A. is monotonically increasing
• B. is monotonically decreasing
• C. has a point of maxima
• D. has a point of minima

Answer 1

The correct option is A is monotonically increasing

Here, $f\left(x\right)=4\tan x-{\tan }^{2}x+{\tan }^{3}x$
$\Rightarrow f^{\prime }\left(x\right)=4{\sec }^{2}x-2\tan x{\sec }^{2}x+3{\tan }^{2}x{\sec }^{2}x$
$={\sec }^{2}x(4-2\tan x+3{\tan }^{2}x)$
$=3{\sec }^{2}x\{{\tan }^{2}x-\frac{2}{3}\tan x+\frac{4}{3}\}$
$=3{\sec }^{2}x\{{(\tan x-\frac{1}{3})}^2+(\frac{4}{3}-\frac{1}{9})\}$
$=3{\sec }^{2}x\{{(\tan x-\frac{1}{3})}^2+\frac{11}{9}\}>0,\forall x$
Therefore, $f\left(x\right)$ is increasing for all $x\in$ domain.