Determine Concavity
To determine concavity, we need to examine the second derivative of the function
To determine concavity, we need to examine the second derivative of the function. The second derivative tells us whether the graph is concave up or concave down.
1. Start with the original function f(x).
2. Calculate the first derivative of f(x) to find f'(x).
3. Calculate the second derivative of f(x) by taking the derivative of f'(x). This will give us f”(x).
4. Analyze the sign of f”(x) to determine concavity:
a. If f”(x) > 0 for all x in the domain, then the graph is concave up.
b. If f”(x) < 0 for all x in the domain, then the graph is concave down. c. If f''(x) changes sign at any point, then the graph has points of inflection where the concavity changes. Let's illustrate this with an example: Example: Determine the concavity of the function f(x) = 3x^2 - 6x + 2. Step 1: Calculate the first derivative of f(x): f'(x) = 6x - 6 Step 2: Calculate the second derivative of f(x): f''(x) = (6x - 6)' = 6 Step 3: Analyze the sign of f''(x): Since f''(x) = 6, which is always positive, the graph is concave up for all values of x. There are no points of inflection since f''(x) doesn't change sign. Therefore, the function f(x) = 3x^2 - 6x + 2 is concave up for all x.
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