Let $f(x)$ be a function defined by $f(x) = 2x^3 - 24x + 107$. If $\alpha$ and $\beta$ are the maximum and minimum values respectively in the interval $[1,3]$, then $\alpha-\beta$ is equal to:

The rate of change of volume of a sphere with respect to its surface area, when the radius is 6cm is:

For the function $f(x)=ax^3-9ax^2+9x+3, x\in R$, then which of the following statements are True?

(A) $f'(x)=3[ax^2-6ax+3]$

(B) It is increasing for $a\in\left[0,\dfrac{1}{3}\right]\ \forall x\in R$

(C) It is increasing for $a\in(-\infty,0)\ \forall x\in R$

(D) It is increasing for $\forall a\in R$

Choose the correct answer from the options given below:

The total cost $C(x)$ in Rupees associated with the production of $x$ units of an item is given by $C(x) = 0.007x^3 + 26x^2 + 15x + 400$. The marginal cost when 10 items are produced is:

The interval in which the function $f(x) = x^2 - 3x + 7$ is increasing, is

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 $(0, \pi)$

Choose the correct answer from the options given below:

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: