Question

Let \( P(x) = x^4 + ax^3 + bx^2 + cx + d \) be such that \( x = 0 \) is the only real root of \( P'(x) = 0 \). If \( P(-1)<P(1) \), then in the interval \( [-1,1] \):

• A. \( P(-1) \) is not minimum of \( P(x) \), but \( P(1) \) is the maximum of \( P(x) \)
• B. \( P(-1) \) is minimum of \( P(x) \), but \( P(1) \) is not the maximum of \( P(x) \)
• C. Neither \( P(-1) \) is the minimum nor \( P(1) \) is the maximum of \( P(x) \)
• D. \( P(-1) \) is the minimum and \( P(1) \) is the maximum of \( P(x) \)

Hint:

For problems involving extrema on intervals, first analyze the derivative roots and verify increasing or decreasing behavior using sign tests.

Answer 1

The Correct Option is D

Since \( P'(x) \) has only one real root at \( x = 0 \), \( P(x) \) has only one critical point at \( x = 0 \). If \( P(-1)<P(1) \), then the function is increasing in \( [-1,1] \), making \( P(-1) \) the minimum and \( P(1) \) the maximum.