Solution:
We have, $\text{f(x)}=4\sin^3\text{x}-6\sin^2\text{x}+12\sin\text{x}+100$
$\therefore\ \text{f(x)}=12\sin^2\text{x}\cdot\cos\text{x}-12\sin\text{x}\cdot\cos\text{x}+12\cos\text{x}$
$=12\cos\text{x}\big[\sin^2\text{x}-\sin\text{x}+1\big]$
$=12\cos\text{x}\big[\sin^2\text{x}+(1-\sin\text{x}\big]$
Now, $1-\sin\text{x}\geq0\text{ and }\sin^2\text{x}\geq0$
$\therefore\ \sin^2\text{x}+\text{f}-\sin\text{x}>0$
Hence, f'(x) > 0, when $\cos\text{x}>0\text{ i.e., x}\in\Big(\frac{-\pi}{2},\frac{\pi}{2}\Big)$
So, f(x) is increasing when $\text{x}\in\Big(-\frac{\pi}{2},\frac{\pi}{2}\Big)$
and f'(x) < 0, when $\cos\text{x}<0\text{ i.e., x}\in\Big(\frac{\pi}{2},\frac{3\pi}{2}\Big)$
Hence, f(x) is decreasing when $\text{x}\in\Big(\frac{\pi}{2},\frac{3\pi}{2}\Big)$
Hence, f(x) is decreasing in $\Big(\frac{\pi}{2},\pi\Big)$
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
$[A]$ $f(x)$ is increasing in $(0, \infty)$
$[B]$ $f(x)$ is decreasing in $(0, \infty)$
$[C]$ $f(x)>e^{2 x}$ in $(0, \infty)$
$[D]$ $f^{\prime}(x) < e^{2 x}$ in $(0, \infty)$
$x+y+z=5, x+2 y+\lambda^2 z=9$
$x+3 y+\lambda z=\mu$, where $\lambda, \mu \in R$. Then, which of the following statement is NOT correct?