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STD 12 - 6. Application of derivatives — Mathematics STD 12 Science — Question

CBSE BoardEnglish MediumSTD 12 ScienceMathematicsSTD 12 - 6. Application of derivatives1 Mark
MCQ
Let $f(x)$ be a non-negative differentiable function on $[0, \infty)$ such that $f(0)=0$ and $f^{\prime}(x) \leq 2 f(x)$ for all $x>0$. Then, on $[0, \infty)$.
  • ✓
    $f(x)$ is always a constant function
  • B
    $f(x)$ is strictly increasing
  • C
    $f(x)$ is strictly decreasing
  • D
    $f^{\prime}(x)$ changes sign
✓

Answer

Correct option: A.
$f(x)$ is always a constant function
a
(a)

We have

$f(x)$ is non-negative differentiable function on $[0, \infty)$

$f(0) =0$

$f^{\prime}(x) \leq 2 f(x)$

$f^{\prime}(x) \leq 2 f(x)$

$\log f(x) \leq 2 x+c$

$f(x) \leq A e^{2 x}$

$f(0) \leq A$

$A =0 \quad \quad \mid f(0)=0 \text { and } f(x) \geq 0]$

$f(x) =0$

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