The maximum slope of the tangents to the curve $y(x) = -x^3 + 3x^2 + 9x - 30$ is

Let $f(x) = \log_e(\sin x), x \in (0, \pi)$, then which of the following statements is/are TRUE?

(A) $f(x)$ is increasing on $(0, \pi/2)$

(B) $f(x)$ is decreasing on $(\pi/2, \pi)$

(C) $f(x)$ is increasing on $(0, \pi)$

(D) $f(x)$ is decreasing on $(0, \pi)$

Choose the correct answer from the options given below:

$f(x)=\sin x+\frac{1}{2} \cos 2 x \text { in }\left[0, \frac{\pi}{2}\right]$

(A) $f^{\prime}(x)=\cos x-\sin 2 x$

(B) The critical points of the function are $x=\frac{\pi}{6}$ and $x=\frac{\pi}{2}$

(C) The minimum value of the function is $2$

(D) The maximum value of the function is $\frac{3}{4}$

Choose the correct answer from the options given below :

Match List-I with List-II

Choose the correct answer from the options given below:

The slope of normal to the curve $y = 3x^2 + 3 \sin 3x$, at $x = 0$ is:

The interval in which the function $f(x) = 10 - 6x - 2x^2$ is decreasing is :

The function $f(x)=81-3x^3$ has