Question

The condition for $y=a{x}^4+b{x}^3+c{x}^2+dx+e$ to have points of inflection is

• A. $b^2-4ac>0$
• B. $3b^2-8ac=0$
• C. $3b^2-8ac>0$
• D. $3b^2-8ac<0$

Answer 1

The correct option is C $3b^2-8ac>0$

$f^{\prime \prime }\left(x\right)=0$ for inflection point.
$f^{\prime }\left(x\right)=4a{x}^3+3b{x}^2+2cx$
$f^{\prime \prime }\left(x\right)=12a{x}^2+6bx+2c=0$
Roots of this equation will be
$x=\frac{-3b\pm \sqrt{9b^2-24ac}}{12a}$
So $9b^2-24ac>0$
$3b^2-8ac>0$