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:

The function $f: R \rightarrow R$ (where $R$ is set of real numbers) defined as

$f(x) = x^2 + 2x$ is

The function $f:R\to R$ is defined by $f(x)=2x^3+5$, then which of the following statements are correct ?

(A) $f(x)$ has no local maximum value.

(B) $f(x)$ has no local minimum value.

(C) $f(x)$ has both local maximum and local minimum values.

(D) $f(x)$ has neither a local maximum value nor a local minimum value

Choose the correct answer from the options given below:

If $x$ is real, the minimum value of $x^2 - 8x + 20$ is

If a revenue function is given by $R(x) = 2027x - 1013x^2 - 675x^3$, then the marginal revenue function (MR) is:

The rate of change of $\sqrt{x^2-2}$ with respect to $\dfrac{x}{x+1}$ at $x=2$ is :

A cylinder with height 'h' and radius of base is 'r' which is open at the top having a given outer surface area, then volume of the cylinder is maximum if