Question

Let $f(x)=5 x \tan x+8 \sin (\tan x)+5 \ln (\cos x), x \in\left(\frac{-\pi}{4}, 0\right)$ then

• A. $f(x)$ is strictly increasing in $\left(\frac{-\pi}{4}, 0\right)$.
• B. $f ( x )$ has a point of local maximum.
• C. the equation $f ( x )=0$ has a root in $\left(\frac{-\pi}{4}, 0\right)$
• D. $ f(x)<0$ for all $x \in\left(\frac{-\pi}{4}, 0\right)$.

Answer 1

Answer: (A), (D)

$: f(x)=5 x \tan x+8 \sin (\tan x)+5 \ln (\cos x), x \in\left(\frac{-\pi}{4}, 0\right) $
$\Rightarrow f^{\prime}(x)=\sec ^2 x(5 x+8 \cos (\tan x))$
For $x \in\left(\frac{-\pi}{4}, 0\right), 5 x \in(-3.9,0)$ and $8 \cos (\tan x)>4$
Hence, $f^{\prime}(x)>0$ for $\left.x \in\left(\frac{-\pi}{4}, 0\right) \right]$