If it is given that at $x = 1$, the function $f(x) = x^4 - 62x^2 + 2ax + b$ attains its maximum value on the interval [0, 2], then the value of a is:

For the function $f(x) = x^5 - 5x^4 + 5x^3 - 1$, which of the following are true?

(A) $f(x)$ has local minimum value equal to -28.

(B) $f(x)$ has point of inflection at $x=1$.

(C) $f(x)$ has point of local maxima at $x=1$.

(D) $f(x)$ has the local maximum value equal to 2.

Choose the correct answer from the options given below:

The points on the ellipse $4x^2 + y^2 = 4$ that are farthest from the point (1,0) are;

The interval for which the function $f(x)=\frac{x}{x^2+x+1}$ is an increasing function, is:

The volume of a cube is increasing at the rate of 27 cm$^3$/s. How fast is the surface area increasing when the length of the cube is 12 cm.

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.

Choose the correct answer from the options given below:

A car is moving along the curve $y = x^3 + 12$. The point(s) on the curve at which the rate of change of its y-coordinate at a certain time is 3 times the rate of change of its x-coordinate is/are