Question

A cubic polynomial with real coefficients

• A. Can possibly have no extrema and no zero crossings
• B. May have up to three extrema and upto $2$ zero crossings
• C. Cannot have more than two extrema and more than three zero corssings
• D. Will always have an equal number of extrema and zero crossings

Answer 1

The correct option is C Cannot have more than two extrema and more than three zero corssings



$y=a{x}^3+b{x}^2+cx+d$
Let $a>0$
$y(x\to \infty )\to \infty$
$y(x\to -\infty )\to -\infty$
$\frac{dy}{dx}=0$, gives two roots.
So it can have two extremum.
Hence A cubic polynomial with real coefficients can have maximum two extremum and three zero crossings