The interval in which the function $f(x) = x^3-3x^2+3x-20$ is increasing, is

The rate of change in area of a triangle having sides 10cm and 12cm when the variable angle between them at an angle $60°$, is:

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 :-

Let $C(x)$ and $R(x)$ be the cost function and the revenue function for $x$ units of a commodity such that marginal cost of producing 2 units is equal to marginal revenue of selling 1 unit. If $C(x) = ax^3 + bx^2 - 12x + 6$ and $R(x) = 100 + 6ax - bx^2$, then which one of the following is TRUE?

For $f(x) = x^3 - 6x^2 + 9x + 15$, which of the following statements are TRUE?

(A) $f'(x) = 3x^2 - 12x + 15$.

(B) The critical points are 3 and 1.

(C) $x = 1$ is a point of local maxima.

(D) Local minimum value is 19.

Choose the correct answer from the options given below:

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

(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 $\frac{3}{4}$

Choose the correct answer from the options given below :

The point on the curve $y^2 = 8x$ for which the abscissa and ordinate change at the same rate, is