Question

The function $f(x)=\tan x-4 x$ is strictly decreasing on

• A. $\left(-\frac{\pi}{3}, \frac{\pi}{3}\right)$
• B. $\left(\frac{\pi}{3}, \frac{\pi}{2}\right)$
• C. $\left(-\frac{\pi}{3}, \frac{\pi}{2}\right)$
• D. $\left(\frac{\pi}{2}, \pi\right)$

Answer 1

Answer: (A)

$f(x)=\tan x-4 x \Rightarrow f^{\prime}(x)=\sec ^2 x-4$
When $ \frac{-\pi}{3} < x < \frac{\pi}{3}, 1 < \sec x < 2$
Therefore, $ 1 < \sec ^2 x < 4$
$\Rightarrow -3<\left(\sec ^2 x-4\right)<0$
Thus, for $\frac{-\pi}{3}, x<\frac{\pi}{3}, f^{\prime}(x)<0$
Hence, $f$ is strictly decreasing on $\left(\frac{-\pi}{3}, \frac{\pi}{3}\right)$.