Question

Suppose that $f$ is a polynomial of degree $3$ and that $f^{\prime \prime }\left(x\right)\ne 0$ 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 $2$ stationary point
• D. $f$ has exactly $0$ or $2$ stationary point.

Answer 1

The correct option is D $f$ has exactly $0$ or $2$ stationary point.

The given information is a polynomial of degree $3$.

Let the function be $f\left(x\right)=a{x}^3+b{x}^2+cx+d$.

To find the stationary points we differentiate and equate it to zero.

$f^{\prime }\left(x\right)=3a{x}^2+2bx+x$

The differentiated function is a quadratic equation which will give two real or two complex roots. If we get two real roots then we will have exactly $2$ stationary points. Now if the roots are complex as they always occur in pair then we have no stationary points. Thus f has exactly $0$ or $2$ stationary points. .....Answer