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  4. Suppose f (x ) is a polynomial of degree four, having critical points at -1, 0, 1 If T = x ∈ RI f ( x )= f (0) then the sum of squares of all the elements of T is:

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Q. Suppose $f (x )$ is a polynomial of degree four, having critical points at $-1, 0, 1 $ If $T =\{ x \in RI f ( x )= f (0)\}$, then the sum of squares of all the elements of $T$ is:

JEE MainJEE Main 2020Application of Derivatives

Solution:

$f^{\prime}(x)=x(x+1)(x-1)=x^{3}-x$
$\int d f(x)=\int x^{3}-x d x$
$f(x)=\frac{x^{4}}{4}-\frac{x^{2}}{2}+C$
$f(x)=f(0)$
$\frac{x^{4}}{4}-\frac{x^{2}}{2}=0$
$x^{2}\left(x^{2}-2\right)=0$
$x=0,0, \sqrt{2},-\sqrt{2}$
$x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=0+2+2=4$