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

A solid sphere of gold of radius 5 cm, is being melted in a furnace such that the radius is decreasing uniformly. When the radius is 1cm, the rate at which surface area (A) is decreasing w.r.t. time ($t$) is equal to

($\frac{dv}{dt}$ = rate at which volume is decreasing w.r.t. $t$)

A spherical ice ball is melting at the rate of 100 $\pi$ cm³/min. The rate at which its radius is decreasing when its radius is 15 cm, is

If the total cost function is given by $C(x)=\frac{x^3}{3}-2x^2+9x-70$ and the selling price per unit is ₹6, then for what value of $x$ will the profit be maximum?

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:

If $f(x) = (x+3)e^{-x}$, then which of the following statements are True?

(A) Function $f(x)$ is increasing on $(-2, \infty)$.

(B) Function $f(x)$ is decreasing on $(-2, \infty)$.

(C) Function $f(x)$ is increasing on $(-\infty, -2)$.

(D) Function $f(x)$ is decreasing on $(2, 3)$.

Let $f(x) = 4x^3 - 18x^2 + 27x - 5$, $x \in R$. Then which of the following statements are TRUE?

(A) $f''(x) = 24x - 36$

(B) f has local maxima at $x = \frac{3}{2}$ but no minima

(C) f has neither maxima nor minima

(D) f has both maxima and minima

Choose the correct answer from the options given below: