The function $f(x)=17-9x+6x^2-x^3$ is strictly increasing in the interval, is:

$f(x)=\sin x+\frac{1}{2} \cos 2 x \text { in }\left[0, \frac{\pi}{2}\right]$

(A) $f^{\prime}(x)=\cos x-\sin 2 x$

(B) The critical points of the function are $x=\frac{\pi}{6}$ and $x=\frac{\pi}{2}$

(C) The minimum value of the function is $2$

(D) The maximum value of the function is $\frac{3}{4}$

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A particle moves along a curve $3y^2 = 6x^3 + 8$ such that its y-coordinate is changing at the rate of 9 cm/sec at point (1,2). Then the rate of a change of x-coordinate at the same point 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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If the minimum value of $a$ is $-\frac{k}{2}$ such that the function $f(x) = x^2 + ax + 5$ is increasing in [1, 2]. Then value of $k$ 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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If the lengths of the three sides of a trapezium other than the base are $10 \mathrm{~cm}$ each, then the maximum area of the trapezium is: