The two positive numbers whose sum is 16 and the sum of whose squares is minimum then the positive numbers are:

Consider the function $f(x) = x^3 - 3x$. Then

Match List-I with List-II

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For $x > 0$, the minimum value of $\frac{x}{\log_e x}$ is

Consider the function $f(x) = -2x^3 + 3x^2 + 36x - 15$, then which of the following are TRUE?

A. $f(x)$ increases on $(-2, 3)$

B. $f(x)$ decreases on $(-\infty, -2) \cup (3, \infty)$

C. $f(x)$ increases on $(-\infty, -2) \cup (3, \infty)$

D. $f(x)$ neither increases nor decreases on $(-2, 3)$

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The function $f(x) = \sin 3x$, $x \in \left[0, \frac{\pi}{2}\right]$

(A) is increasing on $\left[0, \frac{\pi}{6}\right]$

(B) is decreasing on $\left[\frac{\pi}{6}, \frac{\pi}{2}\right]$

(C) is increasing on $\left[0, \frac{\pi}{2}\right]$

(D) is decreasing on $\left[0, \frac{\pi}{2}\right]$

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An energy DRONE is flying along the curve $y = x^2 + 7$. A soldier is placed at $(3, 7)$. The nearest distance of the DRONE from soldier's position is

The function $f(x) = 4x^3 - 7x^2$ has point(s) of local minima at