If $f(x) = x^2 - 4x + 13, x \in \mathbb{R}$, then which of the following are correct?

(A) $x = 2$ is a stationary point of $f(x)$.

(B) $f(x)$ is increasing function on $(2, \infty)$

(C) $f(x)$ have maxima at $x = 2$

(D) $f(2) = 9$

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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:

The cost function of a product is given by $C(x)=\frac{1}{3}x^3-45x^2-900x+36$, where x is the number of units produced. In order to minimize the marginal cost, how many units of the product must be produced?

Let $f(x) = \log_e(\sin x), x \in (0, \pi)$, then which of the following statements is/are TRUE?

(A) $f(x)$ is increasing on $(0, \pi/2)$

(B) $f(x)$ is decreasing on $(\pi/2, \pi)$

(C) $f(x)$ is increasing on $(0, \pi)$

(D) $f(x)$ is decreasing on $(0, \pi)$

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For the function $f(x) = sinx + cosx, x \in [0, \pi]$, which one of the following is correct?

The point of local maxima of the function $f(x) = (x - 2)^5(x + 2)^2$ is

Rate of change of area of a circle with respect to radius

(A) When r = 2 cm is $4\pi$ cm²/cm.

(B) When r = 6 cm is $12\pi$ cm²/cm.

(C) When r = 12 cm is $6\pi$ cm²/cm.

(D) When r = 16 cm is $8\pi$ cm²/cm.

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