JEE Advanced 2017 · Paper 2 · Mathematics Q10

MathematicsMultiple correct+4 / -2Question 46 of 54

If $f: \mathbb{R} \to \mathbb{R}$ is a differentiable function such that $f'(x) > 2f(x)$ for all $x \in \mathbb{R}$ , and $f(0) = 1$ , then

f(x)

is increasing in

(0, \infty)

f(x)

is decreasing in

(0, \infty)

f(x) > e^{2x}

(0, \infty)

f'(x) < e^{2x}

(0, \infty)

Answer and worked solution

Official answer: A, C

Integrating factor. Let $h(x)=e^{-2x}f(x)$ . Then
$h'(x)=e^{-2x}(f'-2f)>0,$
so $h$ is increasing. Since $h(0)=1$ , $h(x)>1$ for $x>0$ .

Consequences for x > 0.

(A), (C)

Topics:Linear first-order equations Monotonicity

JEE Advanced 2017 · Paper 2, Mathematics question 10