Question

Consider a real valued continuous function $f ( x )$ defined on the interval $[ a , b ]$. Which of the following statements does not hold(s) good?

• A. If $f(x) \geq 0$ on $[a, b]$ then $\int\limits_a^b f(x) d x \leq \int\limits_a^b f^2(x) d x$.
• B. If $f(x)$ is increasing on $[a, b]$, then $f^2(x)$ is increasing on $[a, b]$.
• C. If $f(x)$ is increasing on $[a, b]$, then $f(x) \geq 0$ on $(a, b)$.
• D. If $f ( x )$ attains a minimum at $x = c$ where $a < c < b$, then $f ^{\prime}( c )=0$.

Answer 1

Answer: (A), (B), (C), (D)

(A) may be false since if $0< f(x)< 1, f^2(x)< f(x)$ and $\int\limits_a^b f(x) d x >\int\limits_a^b f^2(x) d x$
(B) may be false since if $f(x)<0, \frac{d}{d x} f^2(x)=2 f(x) f^{\prime}(x)<0$ when $f^{\prime}(x)>0$ and so $f^2(x)$ is decreasing while $f ( x )$ is increasing
(C) may be false since a function can be negative and increasing.
(D) may be false since a function may not be differentiable at $x = c$ for which it attains its minimum.