Let f(x) be a non-negative differentiable function on [0, ∞) such that f (0)=0 and f'(x) le 2f(x) for all x 0 Then, on [0, ∞).

Question

Let f(x) be a non-negative differentiable function on [0, ∞) such that f (0)=0 and f'(x) le 2f(x) for all x> 0 Then, on [0, ∞). (A) f (x) is always a constant function (B) f (x) is strictly increasing (C) f (x) is strictly decreasing (D) f'(x) changes sign

Tardigrade Admin · Accepted Answer

We have, f (x) is non-negative differentiable function on [0, ∞) f(0)=0, f' (x) le 2f (x) ⇒ f'(x) le 2 f (x) ⇒ log f (x) le 2x+c ⇒ f (x) le Ae2x ⇒ f (0) le A ⇒ A=0 [f (0)=0 and f (x) le 0] ∴ f (x)=0 Hence, f (x) is always a constant function

Q. Let $f (x)$ be a non-negative differentiable function on $[0, \infty)$ such that $f (0) = 0$ and $f^{'} (x) \leq 2 f (x)$ for all $x > 0$ Then, on $[0, \infty)$ .

$f (x)$ is always a constant function

$f (x)$ is strictly increasing

$f (x)$ is strictly decreasing

$f^{'} (x)$ changes sign

Q. Let f(x) be a non-negative differentiable function on [0,∞) such that f(0)=0 and f′(x)≤2f(x) for all x>0 Then, on [0,∞).

f(x) is always a constant function

f(x) is strictly increasing

f(x) is strictly decreasing

f′(x) changes sign

We have, f(x) is non-negative differentiable function on [0,∞) f(0)=0,f′(x)≤2f(x) ⇒f′(x)≤2f(x) ⇒logf(x)≤2x+c ⇒f(x)≤Ae2x ⇒f(0)≤A ⇒A=0 [f(0)=0 and f(x)≤0] ∴f(x)=0 Hence, f(x) is always a constant function

Q. Let $f (x)$ be a non-negative differentiable function on $[0, \infty)$ such that $f (0) = 0$ and $f^{'} (x) \leq 2 f (x)$ for all $x > 0$ Then, on $[0, \infty)$ .

$f (x)$ is always a constant function

$f (x)$ is strictly increasing

$f (x)$ is strictly decreasing

$f^{'} (x)$ changes sign