GATE CS 2016 Set 2 Q02 — Calculus: Limits, Continuity, Differentiability

Any polynomial $f(x)$ can be uniquely decomposed into its even part $P(x)$ and odd part $Q(x)$:

where $P(x) = \frac{f(x) + f(-x)}{2}$ is an even polynomial, and $Q(x) = \frac{f(x) - f(-x)}{2}$ is an odd polynomial.

We are given that the degree of $f(x) + f(-x)$ is $10$. Since $f(x) + f(-x) = 2P(x)$, the degree of the even polynomial $P(x)$ is $10$.

Now, let us find the derivative $g(x) = f'(x)$:

Using the properties of derivatives of even and odd functions:

• Since $P(x)$ is even, its derivative $P'(x)$ is an odd function, meaning $P'(-x) = -P'(x)$.
• Since $Q(x)$ is odd, its derivative $Q'(x)$ is an even function, meaning $Q'(-x) = Q'(x)$.

Thus, we can compute $g(x) - g(-x)$ as follows:

Since $P(x)$ is a polynomial of degree $10$, its derivative $P'(x)$ is a polynomial of degree $10 - 1 = 9$. Therefore, the degree of $g(x) - g(-x)$ is 9.