Question

A differentiable function $y=f(x)$ is such that its graph cuts the curve $y=a x^2+b x+c$ at 10 distinct points. Find the minimum number of distinct roots of $f ^{\prime \prime \prime}( x )=0$.

Answer 1

$O$ MR Let $g(x)=f(x)-\left(a x^2+b x+c\right)$.
$\text { Since, } g(x)=0 \text { has } 10 \text { distinct real roots, } $
$\Rightarrow g^{\prime}(x)=0 \text { has minimum } 9 \text { distinct real roots, }$
$\Rightarrow g^{\prime \prime}(x)=0 \text { has minimum } 8 \text { distinct real roots, } $
$\Rightarrow g^{\prime \prime \prime}(x)=0 \text { has minimum } 7 \text { distinct real roots, } $
$\Rightarrow f ^{\prime \prime \prime}(x)=0 \text { has atleast } 7 \text { distinct real roots. Ar }$