Question

At $x=0$, the function $f\left(x\right)=x^3+1$ has

• A. a maximum value
• B. a minimum value
• C. a singularity
• D. a point of inflection

Answer 1

The correct option is D a point of inflection

Given, $f\left(x\right)=x^3+1$
$\Rightarrow f^{\prime }\left(x\right)=3x^2=0$
$\Rightarrow x=0$ is a point of maxima or minima.
Now, $f^{\prime \prime }\left(x\right)=6x$
$\Rightarrow f^{\prime \prime }\left(0\right)=0$
$\Rightarrow f^{\prime \prime \prime }\left(x\right)=6$
$\because f^{\prime }\left(x\right)$ and $f^{\prime \prime }\left(x\right)$ is zero at $x=0$ and $f^{\prime \prime \prime }\left(x\right)$ is positive hence it is a piont of inflection.