If-f-x-is-an-even-function-and-satisfies-the-relation-x-2-f-x-2f-x-g-x-where-g-x-is-an-odd-function-then-the-value-of-f-5-is-a-0-b-37-75-c-4-d-51-77-

Question Number 128424 by bramlexs22 last updated on 07/Jan/21

If f (x) is an even function and

satisfies the relation x^{2} f (x) - 2 f (x) = g (x)

where g (x) is an odd function

then the value of f (5) is

(a) 0 (b) \frac{37}{75} (c) 4 (d) \frac{51}{77}

Answered by mr W last updated on 07/Jan/21

x^{2} f (x) - 2 f (x) = g (x) \dots (i)

{(- x)}^{2} f (- x) - 2 f (- x) = g (- x)

x^{2} f (x) - 2 f (x) = - g (x) \dots (i i)

(i) + (i i) :

2 (x^{2} f (x) - 2 f (x)) = 0

(x^{2} - 2) f (x) = 0

(5^{2} - 2) f (5) = 0

\Rightarrow f (5) = 0

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