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Q.
Suppose that $f$ is a polynomial of degree $3$ and that $f''(x) \neq 0$ at any of the stationary point. Then
Application of Derivatives
Solution:
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 $f'(x)=m(x-\alpha)^{2}$, when $\alpha$ is the repeated roots and $m \in R$.
So, our original function $f$ has a critical point at $x=\alpha$.
Also, $f ^{\prime \prime}( x )=2 m ( x -\alpha)$, in which case $f ''(\alpha)=0$.
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.