Let $C(x)$ and $R(x)$ be the cost function and the revenue function for $x$ units of a commodity such that marginal cost of producing 2 units is equal to marginal revenue of selling 1 unit. If $C(x) = ax^3 + bx^2 - 12x + 6$ and $R(x) = 100 + 6ax - bx^2$, then which one of the following is TRUE?

Match List I with List II

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For the function $f(x) = e^x + e^{-x}$

(A) $f'(x) = e^x - e^{-x}$

(B) The critical point is $x = 0$

(C) The minimum value is 1

(D) $x = 0$ is the point of local minimum.

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The demand function for a certain product is represented by $p = 35 + 5x - 3x^2$, where $x$ is the number of units demanded and $p$ is the price per unit. The marginal revenue when 2 units are sold is

The length of a rectangle is decreasing at the rate of 4 cm/minute and the width is increasing at the rate of 3 cm/minute, then the rate of change of the perimeter is

If the maximum value of the function $f(x) = \frac{\log_ex}{x}$, $x > 0$ occurs at $x = a$, then $a^2f''(a)$ is equal to

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.

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