First slide

Maxima and minima

Question

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

Moderate

A

8
B

2
C

4
D

6

Solution

$Given f (x) is a polynomial of degree 4 and having critical points at - 1, 0, 1$
Hence,
$f^{1} (x) = λ x (x^{2} - 1) = λ (x^{3} - x) (λ \neq 0)$
Integrate
$\begin{matrix} f (x) = λ (\frac{x^{4}}{4} - \frac{x^{2}}{2}) + k \\ = λ (\frac{x^{4}}{4} - \frac{x^{2}}{2}) + f (0) \end{matrix}$
$If f (x) = f (0) \Rightarrow λ (\frac{x^{4}}{4} - \frac{x^{2}}{2}) = 0 \Rightarrow x^{2} = 0 or x^{2} = 2 \Rightarrow x = 0, x = \pm \sqrt{2}$
Sum of the squares of the values of x in T is 0 + 2 + 2 = 4.

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