Question

Find the greatest & the least values of the following functions in the given interval, if they exist. $y=2\tan x-{\tan }^{2}x[0,\frac{\pi }{2})$

• A. The greatest value is equal to $1$, no least value
• B. The greatest value is equal to $2$, no least value
• C. No greatest value, least value is $-1$
• D. The greatest value is equal to 1, least value $-2$

Answer 1

Answer: (A)

The greatest value is equal to $1$, no least value

Given, $y=2\tan x-{\tan }^{2}x$

$y^{\prime }\left(x\right)$ $=2{\sec }^2\left(x\right)-2\tan x.{\sec }^{2}x$
$=2{\sec }^2\left(x\right)(1-\tan x)$
$=0$
$\tan x=1$ and $\sec x=0$
$x=\frac{\pi }{2}$
And $x=\frac{\pi }{4}$
Hence, $y^{\prime \prime }\left(x\right)$
$=4\sec \left(x\right).\sec \left(x\right)\left(\tan x\right)(1-\tan x)-2{\sec }^{2}x\left({\sec }^2\right(x\left)\right)$
$={\sec }^2\left(x\right)[4\tan x-4{\tan }^{2}x-{\sec }^2(x\left)\right]$
Hence, $y"\left(\frac{\pi }{4}\right)<0$
Hence, it attains a maximum at $x=\frac{\pi }{4}$
$\therefore y\left(\frac{\pi }{4}\right)=1$.