The function $f(x) = \log_e(\sin x), x \in (0, \pi)$ is (A) strictly increasing on $\left(0, \frac{\pi}{2}\right)$ (B) strictly decreasing on $\left(0, \frac{\pi}{2}\right)$ (C) strictly increasing on $\left(\frac{\pi}{2}, \pi\right)$ (D) strictly decreasing on $\left(\frac{\pi}{2}, \pi\right)$ (E) strictly increasing on $(0, \pi)$ Choose the correct answer from the options given below:

A function $f(x)=\sin x(1+\cos x), x \in [0,\pi]$ has a local maxima at $x=$

A steel plant is capable of producing 'x' tonnes per day of low-grade steel and 'y' tonnes per day of high-grade steel, where $y=\frac{32-5x}{10-x}$. If the fixed price of low-grade steel is half that of high-grade steel, then the quantity of low-grade steel that should be produced per day for maximum receipts is

An open box is made from a cardboard measuring 12cm x 12cm by cutting off equal squares from the corner and turning up the side. Then maximum possible volume of the box would be:

Match List-I with List-II

Choose the correct answer from the options given below:

For $x \in \mathbb{R}$, if $f(x) = -(x-1)^2 + 2$, then

(A) $f$ is an increasing function on $(-\infty, 1]$

(B) $f$ has no critical points

(C) $f$ has a maximum value at $x = 1$

(D) $f$ has a minimum value at $x = 1$

Choose the correct answer from the options given below:

The maximum slope of the curve $y = -x^3 + 3x^2 + 9x - 27$ is: