GATE | GATE-CS-2016 (Set 2) | Question 12

Let f (x) be a polynomial and g(x) = f'(x) be its derivative. If the degree of (f(x) + f(−x)) is 10, then the degree of (g(x) − g(−x)) is ___________ .

[This Question was originally a Fill-in-the-blanks Question]
(A) 8
(B) 7
(C) 9
(D) 10


Answer: (C)

Explanation: f(x) can be either an even function or an odd function.
If f(x) was an odd function, then f(x) + f(-x) = 0, but here it has been given that it has degree 10. So, it must be an even function.
Therefore, f(x) = f(-x)
=> f'(x) = -f'(-x)
Also, it has been mentioned that g(x) is the derivative of f(x).
So, g(x) = f'(x) and g(-x) = -f'(-x)
=> g(x) – g(-x) = f'(x) – (-f'(-x))
=> g(x) – g(-x) = f'(x) + f'(x)
=> g(x) – g(-x) = 2 * f'(x)
But, f'(x) will have degree 9.

Thus, (C) is the correct option.

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