First slide

Differentiability

Question

$Let f : R \to R is a polynomial function which satisfies$ $f (x + y) + f (x - y) = 2 f (x) - 2 y^{2} \forall x, y \in R, f^{'} (0) \neq 0 and f (0) = 0 . Then which of$ the following is/are true

Difficult

A

$f (x) is always even function$
B

$f^{11} (2) = - 2$
C

$Area bounded by y = f (x) and x -axis is |\frac{{(f^{1} (0))}^{3}}{6}|$
D

$Area bounded by y = f (x) and x -axis is |\frac{{(f^{1} (0))}^{3}}{12}|$

Solution

$\begin{array}{l} p u t x = 0 i n t h e g i v e n e q u a t i o n t h e n f (y) + f (- y) = - 2 y^{2} (\sin c e f (0) = 0) and \\ p u t y = x i n t h e g i v e n e q u a t i o n t h e n f (2 x) - 2 f (x) = - 2 x^{2} \\ f (x) must be quadratic \\ f (x) = - x^{2} + λ x where λ = f^{1} (0) \\ area = \int_{0}^{λ} (- x^{2} + λ x) d x = \frac{λ^{3}}{6} \end{array}$

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