Question

Let $f(x)=e^{|x^2-4x+3|}$ 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 $x=1,3$.