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

Correct Answer is : ∫01f(x)2dx≤100

\int^{1}_{0} \frac{f(x)}{(1+x)^2} dx\leq10

-10\leq\int^{1}_{0}\sin(100x)f(x)dx\leq10

KVPY SX SB Exam 20141 Question Paper

Question : 12 of 120

Marks: +1, -0

Let

f : [0 , 1 ] →[0,∞)

be a continuous function such that

∫↖{1}↙{0}f(x)d x = 10

. Which of the following statements is NOT necessarily true?

Solution:

∵ f(x) ≥ 0

∫↖{1}↙{0}f(x)^2d x ≤ 100

not necessarily true.
because

(\f(x))^2

can take very high values then area bounded by

(f(x))_2

, x-axis & x = 0 to 1 may cross 100.

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