Function $f(x) = x^x, x > 0$ decreases on the interval

The largest interval, in which the function $f(x) = x^3 + 2x^2 - 1$ is increasing, is:

The interval(s), where the function $f(x) = \begin{cases} \frac{1-e^x}{e^{2x}-1} & : x \neq 0 \\ \frac{-1}{2} & : x = 0 \end{cases}$ is increasing, is/ are:

A function $f(x)=\sin x(1+\cos x), x \in [0,\pi]$ has a local maxima at $x=$

For the function, $f(x) = \frac{-3}{4}x^4 - 8x^3 - \frac{45}{2}x^2 - 350$, which of the following statements are correct?

(A) $x = -3$ and $x = -5$ are the only critical points of the given function.

(B) $x = -3$ is a point of local minimum.

(C) The local minimum value at $x = -3$ is 23.1.

(D) $x = -5$ is a point of local maximum.

Choose the correct answer from the options given below:

The point on curve $y^2 = 4x$ which is nearest to the point $(2, -8)$ is

The function $f(x) = 6 - 6x - 2x^2$