Question

Find the interval in which the graph of the function given by \(f(x) = x^4 - 4x^3 + 10\) is concave up.

• A. (-∞, 1) ∪ (3, ∞)
• B. (-∞, 0) ∪ (2, ∞)
• C. (1.5, ∞)
• D. (0, 2)

Hint:

Concave up means the graph is shaped like a cup (U).
Solve \(f''(x) > 0\) to find the intervals.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
Concavity is determined by the second derivative.
If \(f''(x) > 0\), the function is concave up.

Step 2: Key Formula or Approach:
Find \(f''(x)\) and solve \(f''(x) > 0\).

Step 3: Detailed Explanation:
Given \(f(x) = x^4 - 4x^3 + 10\).
First derivative: \[ f'(x) = 4x^3 - 12x^2. \]
Second derivative: \[ f''(x) = 12x^2 - 24x = 12x(x - 2). \]
Set \(f''(x) > 0\): \[ 12x(x - 2) > 0. \]
The critical points are \(x = 0\) and \(x = 2\).
Test intervals: - For \(x < 0\): both \(x\) and \(x-2\) are negative, product is positive, so \(f'' > 0\).
- For \(0 < x < 2\): \(x > 0\), \(x-2 < 0\), product is negative, so \(f'' < 0\).
- For \(x > 2\): both positive, product is positive, so \(f'' > 0\).
Thus, concave up on \((-∞, 0) \cup (2, ∞)\).
This matches option (B).