Question

The number of critical point(s) for $f\left(x\right)=\frac{5}{4}(x-1{)}^{\frac{4}{5}}+x,$ is

• A. More than $2$
• B. $0$
• C. $1$
• D. $2$

Answer 1

The correct option is D $2$

Given : $f\left(x\right)=\frac{5}{4}(x-1{)}^{\frac{4}{5}}+x$
$\Rightarrow f^{\prime }\left(x\right)=\frac{1}{(x-1{)}^{\frac{1}{5}}}+1$
$\Rightarrow f^{\prime }\left(x\right)=\frac{1+(x-1{)}^{\frac{1}{5}}}{(x-1{)}^{\frac{1}{5}}}$
We know, Critical point of a function exists where $f^{\prime }\left(x\right)=0$ or not defined.
$\Rightarrow 1+(x-1{)}^{\frac{1}{5}}=0$ $\Rightarrow x=0$ and
at $x=1,f^{\prime }\left(x\right)$ is not defined.
So, total two critical points.