The function $f(x) = 4 - 3x + 3x^2 - x^3$ is (Here $\mathbb{R}$ is set of real numbers)

If $y = a\log x + bx^2 + x$ has extreme values at $x = 1$ and $x = 2$, then

For the function $f(x)=2x^3-21x^2+36x-100$;

Match the LIST-I with LIST-II

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The points in the interval $[0,2\pi]$, where the function $f(x) = \cos2x + 2\sin x\cos x$, attains its local minima, is/are:

A. $\pi/8$

B. $5\pi/8$

C. $13\pi/8$

D. $11\pi/8$

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If $x$ is real, the minimum value of $x^2 - 8x + 20$ is

The function $f(x) = 24x^4 - 32x^3 - 96x^2 + 7$ on the interval $\left[-\dfrac{3}{2}, 4\right]$ is :

A. increasing in $(-1,0)$

B. decreasing in $(0,4)$

C. increasing in $(2,4)$

D. decreasing in $\left(-\dfrac{3}{2},-1\right)$

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The volume of spherical balloon is increasing at the rate of $4 \text{ cm}^3/ \text{sec}$. The rate of increase of its surface area, when the radius is 3cm will be :-