Question

Let $f(x)=e^{\left|x^2-4 x+3\right|}$ then

• A. $f(x)$ decreases in the interval $(1,2) \cup(3, \infty)$.
• B. $f(x)$ increases in the interval $(-\infty, 1) \cup(2,3)$.
• C. $f ( x )$ has one local maximum point and two local minimum points.
• D. $f(x) $ has one local minimum point and two local maximum points.

Answer 1

Answer: (C)

$ f(x)=e^{g(x)}$
$f ^{\prime}( x )= e ^{ g ( x )} \cdot g ^{\prime}( x )$
$\therefore$ Check the nature of $g(x)$.
$g ( x )$ is increasing $(\uparrow)$ in $(1,2) \cup(3, \infty)$
and decreasing $(\downarrow)$ in $(-\infty, 1) \cup(2,3)$.
$g ( x )$ is continuous $\forall x \in R$.
But non-derivable at $x=1,3$
$g(x)$ has local maximum at $x=2$,
and local minimum at $x=1,3$.