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. $11\pi/8$

Choose the correct answer from the options given below:

The interval in which $f(x)=\frac{x}{\log x}$ is increasing, is:

The maximum slope of the tangents to the curve $y(x) = -x^3 + 3x^2 + 9x - 30$ is

The function $f(x)=\sin^4(x)+\cos^4(x)$, on interval $\left(0,\dfrac{\pi}{2}\right)$, has

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:

If the function $f(x) = x^2 - ax - 2$ is strictly decreasing on $(2, 3)$ then $a$ lies in the interval.

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: