Question

Let the function f satisfy f(x).f’(-x) = f(-x).f’(x) for all x and f(0) = 3. The value of f(x).f(-x) for all x is-
(A) 4
(B) 9
(C) 12
(D) 6

Answer 1

Hint: In this question we will try to eliminate the f’(x) term in the equation given, by integration. After that, we will be able to find the value of f(x).f(-x).

Complete step-by-step answer:

We have been given that-
f(x).f’(-x) = f(-x).f’(x)
f(x).f’(-x) - f(-x).f’(x) = 0
When we look closely at this, we can find that this equation is actually the differential of f(x).f(-x), because they are being differentiated by product rule. Hence,
$\dfrac{\operatorname d\left(\mathrm f\left(\mathrm x\right).\mathrm f\left(-\mathrm x\right)\right)}{\operatorname d\mathrm x}=0$
Integrating both sides by x,
f(x).f(-x) = c
At x = 0
f(0).f(0) = c
But f(0) = 3
c = 9

Hence, f(x).f(-x) = 9

Note: In such types of questions, try to convert the given equation into another form. Check closely what is asked in the question and try to convert it into that form. For instance, here we needed to find the value of f(x).f(-x), so we integrated the given equation to eliminate the differential term from the equation.