Let f:[0,1] arrow[0, ∞) be a continuous function such that ∫_0^1 f(x) d x=10 . Which of the following statements is NOT necessarily true?

Correct answer: (D) ∫_0^1 f(x)^2 d x ≤ 100. I=∫_0^1 f (x^2 ) d x Let x^2=t dx=1/(2 √t) dt I=1 ∫_0^1 f(t) dt/(2 √t) ∵ 1/2 ≤ 1/(2 √t) ∫_0^1 f(t) dt/2 ≤ ∫_0^1 f(t) dt/(2 √t) 5 ≤ l

Q31 / 90 · KVPY 2014 (SB/SX) · Mathematics

Let $f:[0,1] \rightarrow[0, \infty)$ be a continuous function such that $\int_{0}^{1} f(x) d x=10$. Which of the following statements is NOT necessarily true?

A
$\int_{0}^{1} e^{-x} f(x) d x \leq 10$
B
$\int_{0}^{1} \frac{f(x)}{(1+x)^{2}} d x \leq 10$
C
$-10 \leq \int_{0}^{1} \sin (100 x) f(x) d x \leq 10$
Correct answer
D
$\int_{0}^{1} f(x)^{2} d x \leq 100$

Solution

$I=\int_{0}^{1} f\left(x^{2}\right) d x$ Let $x^{2}=t$ $\mathrm{dx}=\frac{1}{2 \sqrt{\mathrm{t}}} \mathrm{dt}$ $\mathrm{I}=1 \int_{0}^{1} \mathrm{f}(\mathrm{t}) \frac{\mathrm{dt}}{2 \sqrt{\mathrm{t}}}$ $\because \frac{1}{2} \leq \frac{1}{2 \sqrt{\mathrm{t}}}$ $\int_{0}^{1} \mathrm{f}(\mathrm{t}) \frac{\mathrm{dt}}{2} \leq \int_{0}^{1} \mathrm{f}(\mathrm{t}) \frac{\mathrm{dt}}{2 \sqrt{\mathrm{t}}} \quad 5 \leq \mathrm{l}$

Hence, the correct answer is option: 'D'.

Let $f:[0,1] \rightarrow[0, \infty)$ be a continuous function such that $\int_{0}^{1} f(x) d x=10$. Which of the following statements is NOT necessarily true?

Solution

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