Question

Let \( f(x) \) be a cubic polynomial with \( f(1) = -10 \), \( f(-1) = 6 \), and has a local minima at \( x = 1 \), and \( f'(x) \) has a local minima at \( x = -1 \). Then \( f(3) \) is equal to _________.

Hint:

For a cubic polynomial, the point where \( f'(x) \) has a local extrema is the same as the point of inflection of \( f(x) \). This point always satisfies \( f''(x) = 0 \).

Answer 1

Correct Answer: 22

Step 1: Understanding the Concept:
We use the properties of derivatives to define the cubic polynomial. A local minimum of \( f(x) \) implies \( f'(x) = 0 \), and a local minimum of \( f'(x) \) implies \( f''(x) = 0 \).
Step 2: Detailed Explanation:
Let \( f'(x) = 3ax^2 + 2bx + c \). Given \( f'(x) \) has a local minima at \( x = -1 \), then \( f''(-1) = 0 \). \( f''(x) = 6ax + 2b \implies 6a(-1) + 2b = 0 \implies b = 3a \).
Given \( f(x) \) has a local minima at \( x = 1 \), then \( f'(1) = 0 \). \( 3a(1)^2 + 2b(1) + c = 0 \implies 3a + 2(3a) + c = 0 \implies c = -9a \).
Now, integrate \( f'(x) \) to find \( f(x) \): \[ f'(x) = 3ax^2 + 6ax - 9a \] \[ f(x) = ax^3 + 3ax^2 - 9ax + d \]
Use given values: 1. \( f(1) = -10 \implies a + 3a - 9a + d = -10 \implies -5a + d = -10 \)
2. \( f(-1) = 6 \implies -a + 3a + 9a + d = 6 \implies 11a + d = 6 \)
Subtracting the equations: \( (11a + d) - (-5a + d) = 6 - (-10) \implies 16a = 16 \implies a = 1 \).
Then \( d = -10 + 5(1) = -5 \).
The polynomial is \( f(x) = x^3 + 3x^2 - 9x - 5 \).
Calculate \( f(3) \): \[ f(3) = 3^3 + 3(3^2) - 9(3) - 5 = 27 + 27 - 27 - 5 = 22 \]
Step 3: Final Answer:
The value of \( f(3) \) is 22.