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

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

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

Which of the following statements are correct ?

A. The function $f(x) = a^x$ is increasing on $(-\infty,\infty)$ if $a>1$.

B. $f(x) = x^2 - 2x$ is decreasing in $(-\infty,1)$.

C. $f(x) = x(x-3)^2$ increases for $1 \le x \le 3$.

D. $f(x) = e^{-x}$ decreases in $[0,\infty)$.

Choose the correct answer from the options given below:

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 maximum and the minimum value of the function $f(x) = -||x+3|-5| + 2$ are :

Let $f(x)$ be a function defined on an interval $[a,b]$ such that $f(x)$ is differentiable on $(a,b)$. Then

Match the LIST-I with LIST-II

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