Letfor all real x and y. If f’(0) exists and equals – 1 and f(0) = 1, then find f(2).Correct answer is '-1'. Can you explain this answer?

Question

Letfor all real x and y. If f&rsquo;(0) exists and equals &ndash; 1 and f(0) = 1, then find f(2).Correct answer is '-1'. Can you explain this answer?

Answer

Analytic Method : since

Replacing x by 2x and y by 0, then

= f’(0) = –1 x ∈ R (given)

Integrating, we get f(x) = –x + c

Putting x = 0, then f(0) = 0 + c = 1 (given)

∴ c = 1

then f(x) = 1 – x

∴ f(2) = 1 – 2 = –1