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Q.
$ f^{\prime \prime}(x)>0$ for all $x \in[-3,4]$, then which of the following are always true?
Application of Derivatives
Solution:
(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 $
$\Rightarrow \exists \text { critical point on }[-3,4]$
