examday

/TMUA

Consider the following theorem:

Let $f$ be a function on the real numbers whose first and second derivatives exist. If $f'(c)=0$ and $f''(c)<0$ then $f$ has a local maximum at $c$.

The following statements, I, II and III, are made.

• I. $f'(x)>0$ is a necessary condition for $f$ to be strictly increasing.
• II. $f(x)$ converges to a finite real number as $x\to\infty$ only if $f'(x)$ converges to zero as $x\to\infty$.
• III. $\int_a^b f(x)\,\mathrm{d}x<\frac12(b-a)(f(a)+f(b))$ is a sufficient condition for $f''(x)>0$ for all $x$ such that $a<x<b$.

Which of the above is/are correct?