Question

Which of the following is always true about a function f(x) on the interval [a, b] ?

• A. If $f\left(x\right)\ge 0$ on [a, b], then ${\int }_a^{b}f\left(x\right)dx\le {\int }_a^b{f}^2\left(x\right)dx$
• B. If $f\left(x\right)$ is increasing on [a, b] then $f^2\left(x\right)$ is increasing on [a, b]
• C. If $f\left(x\right)$ attains a minimum at c where a < c < b, then f ' (c) = 0
• D. None of these

Answer 1

The correct option is D

None of these

(A) May be false if $0<f\left(x\right)<1,f^2\left(x\right)<f\left(x\right)and{\int }_a^b{f}^2\left(x\right)dx\le {\int }_a^{b}f\left(x\right)dx$

(B) May be false if $f\left(x\right)<0,\frac{d}{dx}\left(f^2\right(x\left)\right)=2f\left(x\right)f^{\prime }\left(x\right)<0$ when $f^{\prime }\left(x\right)>0$ and so $f^2\left(x\right)$ is decreasing while $f\left(x\right)$ is increasing.

(C) May be false since a function may not be differentiable at x = c for which it attains its minimum.

Since none of the statements are always true, the correct answer is (D)