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 $f(x)\ge 0$ on $[a,b]$ then $\underset{a}{\overset{b}{\int }}f(x)dx\le \underset{a}{\overset{b}{\int }}{f}^2(x)dx$.
• 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 $f(x)\ge 0$ 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 $\underset{a}{\overset{b}{\int }}f(x)dx>\underset{a}{\overset{b}{\int }}{f}^2(x)dx$
(B) may be false since if $f(x)<0,\frac{d}{dx}{f}^2(x)=2f(x)f^{\prime }(x)<0$ 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.