Answer
See the explanation
Work Step by Step
To determine where the graph of a twice-differentiable function is concave up or concave down, you can use the second derivative test. Here's how it works:
1. Find the Second Derivative: Start by finding the second derivative of the function.
2. Identify Critical Points: Determine the critical points of the function by finding where the first derivative equals zero or is undefined.
3. Test the Sign of the Second Derivative: At each critical point, test the sign of the second derivative:
- If the second derivative is positive, the graph is concave up at that point.
- If the second derivative is negative, the graph is concave down at that point.
- If the second derivative is zero, the test is inconclusive.
4. Analyze the Intervals: Use the signs of the second derivative to determine where the graph is concave up or concave down on intervals between critical points.
Here's an example:
Consider the function \( f(x) = x^3 - 3x^2 + 2x \).
1. Find the Second Derivative:
- First derivative: \( f'(x) = 3x^2 - 6x + 2 \)
- Second derivative: \( f''(x) = 6x - 6 \)
2. Identify Critical Points: Set the first derivative equal to zero and solve for \( x \):
\[ 3x^2 - 6x + 2 = 0 \]
This quadratic equation does not have real roots, so there are no critical points.
3. Test the Sign of the Second Derivative: Since there are no critical points, we can test the sign of the second derivative in different intervals.
- \( f''(x) = 6x - 6 \)
- When \( x < 1 \), \( f''(x) < 0 \), so the graph is concave down.
- When \( x > 1 \), \( f''(x) > 0 \), so the graph is concave up.
So, the graph of \( f(x) \) is concave down for \( x < 1 \) and concave up for \( x > 1 \).
