Question

$f^{\prime \prime }(x)>0$ for all $x\in [-3,4]$, then which of the following are always true?

• A. $f(x)$ has a relative minimum on $(-3,4)$
• B. $f(x)$ has a minimum on $[-3,4]$
• C. $f(x)$ is concave upwards on $[-3,4]$
• D. if $f(3)=f(4)$ then $f(x)$ has a critical point on $[-3,4]$

Answer 1

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

(A) $f(x)$ has no relative minimum on $(-3,4)$
(B) $f(x)$ is continuous function on $[-3,4]$
$\Rightarrow f(x)$ has min. and max. on $[-3,4]$ by IVT
(C) $f^{\prime \prime }(x)>0\Rightarrow f(x)$ is concave upwards on $[-3,4]$

(D) $f(3)=f(4)$
By Rolle's theorem
$\exists c\in (3,4),\text{ where }{f}^{\prime }(c)=0$
$\Rightarrow \exists \text{ critical point on }[-3,4]$