Question

Let $f$ be a polynomial function such that $f'(x)=f(x)\,f''(x)$ for all $x\in\mathbb{R}$. Then:

• A. $f'(2)-f''(2)=0$
• B. $f''(2)-f(2)=4$
• C. $f(2)-f'(2)+f''(2)=10$
• D. $f'(2)+f''(2)=6$

Hint:

When a functional equation involves derivatives of a polynomial, always check constant and linear cases first.

Answer 1

The Correct Option is A

Step 1: Given condition: \[ f'(x)=f(x)\,f''(x) \]
Step 2: Since $f$ is a polynomial, consider its degree. If $\deg f \ge 2$, then $\deg(f') \neq \deg(f\,f'')$ in general, which leads to a contradiction.
Step 3: Hence, the only possible polynomial solution is a constant polynomial. Let \[ f(x)=c \]
Step 4: Then \[ f'(x)=0,\qquad f''(x)=0 \] which satisfies the given condition.
Step 5: Evaluate the given options using \[ f'(2)=0,\quad f''(2)=0 \] \[ f'(2)-f''(2)=0-0=0 \]
Step 6: Hence, option (A) is correct.