Let be a differentiable function on such that and for all

MATHEMATICS · Multi correct · CBSE · Mathematics · Limits and Derivatives · Limits and Derivatives

Problem

Let

f(x)

be a differentiable function on

(-\infty,\infty)

such that

f(1)=5

and

\lim_{a\to x}\frac{af(x)-xf(a)}{a-x}=2

for all

x\in\mathbb{R}

. Which of the following alternatives are correct? (1)

f(x)

has an inflection point. (2)

f'(x)=3,\ \forall x\in\mathbb{R}

(3)

\int_0^2f(x)\,dx=10

(4) The area bounded by

f(x)

and the coordinate axes is

\frac{2}{3}