Question

Suppose that $f$ is a polynomial of degree $3$ and that $f^{\prime \prime }(x)\ne 0$ at any of the stationary point. Then

• A. f has exactly one stationary point
• B. f must have no stationary point
• C. f must have exactly two stationary point
• D. f has either zero or two stationary point

Answer 1

Answer: (D)

The derivative of a degree $3$ polynomial is quadratic.
This must have either $0,1$ or $2$ roots.
If this has precisely one root, then this must be repeated.
Hence, we have $f^{\prime }(x)=m(x-\alpha {)}^2$, when $\alpha$ is the repeated roots and $m\in R$.
So, our original function $f$ has a critical point at $x=\alpha$.
Also, $f^{\prime \prime }(x)=2m(x-\alpha )$, in which case $f^{\prime \prime }(\alpha )=0$.
But we are told that the $2\text{ nd}$ derivative is non-zero at critical point.
Hence, there must be either $0$ or $2$ critical points.