Which of the following statements are correct ?

A. The function $f(x) = a^x$ is increasing on $(-\infty,\infty)$ if $a>1$.

B. $f(x) = x^2 - 2x$ is decreasing in $(-\infty,1)$.

C. $f(x) = x(x-3)^2$ increases for $1 \le x \le 3$.

D. $f(x) = e^{-x}$ decreases in $[0,\infty)$.

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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

For the function $f(x) = \dfrac{3}{2}x^4 - 4x^3 - 45x^2 + 49$. Which of the following statements is/are TRUE ?

A. $f(x)$ is increasing in $(-\infty, -3)$

B. $f(x)$ is increasing in $(-3, 0) \cup (5, \infty)$

C. $f(x)$ is decreasing in $(-\infty, -3) \cup (0, 5)$

D. $f(x)$ is decreasing in $(-\infty, -3) \cup (5, \infty)$

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A function $f(x)$ is defined as:

$f(x) = x^3-6x^2+9x+15$

For the function $f(x)$, which of the following statements are correct ?

(A) The critical points are $x=1$ and $x=3$.

(B) $x=1$ is a point of local minimum.

(C) The local maximum value is 19.

(D) $x=3$ is a point of local minimum.

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If $x = -1$ and $x = -2$ are the extreme points of $f(x) = \alpha\log|x| + \beta x^2 + x$ then

The function $f(x) = 2x^3-15x^2+36x+4$ has local maxima at $x=$

If $x$ is real, then the minimum value of $x^2 - 8x + 25$ is