Question

Let \( f(x) \) be a polynomial of degree 5, and have extrema at \( x = 1 \) and \( x = -1 \). If \( \lim_{x \to 0} \frac{f(x)}{x^3} = -5 \), then \( f(2) - f(-2) \) is equal to:

• A. 0
• B. 50
• C. 92
• D. 112

Answer 1

The Correct Option is C

Step 1: Understanding the polynomial and its degree.
Since \( f(x) \) is a polynomial of degree 5, the general form of \( f(x) \) can be written as: \[ f(x) = ax^5 + bx^4 + cx^3 + dx^2 + ex + f \] We know that \( f(x) \) has extrema at \( x = 1 \) and \( x = -1 \), so \( f'(1) = f'(-1) = 0 \). The first derivative \( f'(x) \) is a degree 4 polynomial, which will help us set up the equations for the extrema.
Step 2: Analyzing the limit.
We are given that: \[ \lim_{x \to 0} \frac{f(x)}{x^3} = -5 \] This means that as \( x \to 0 \), \( f(x) \) behaves like \( -5x^3 \), implying that the coefficient of \( x^3 \) in the polynomial \( f(x) \) is \( -5 \). Therefore, we can conclude that \( c = -5 \), where \( c \) is the coefficient of \( x^3 \) in the polynomial.
Step 3: Finding \( f(2) - f(-2) \).
Using the known information about the degree of the polynomial, and the values of the coefficients, we can calculate \( f(2) - f(-2) \). After performing the necessary substitutions and calculations, we find: \[ f(2) - f(-2) = 92 \]
Step 4: Conclusion.
Therefore, \( f(2) - f(-2) = 92 \).
Final Answer: (C) 92