Explanation
The second derivative of a function indicates how the slope of the function changes. It helps you determine the concavity of a function. If the second derivative is positive, the function is concave up (convex); if it is negative, the function is concave down (concave). You can calculate the second derivative by differentiating the first derivative again.
Procedure
To determine the concavity of a function, follow these steps:
1. Compute the first derivative of the function.
2. Compute the second derivative by differentiating the first derivative again.
3. Check whether the second derivative is positive or negative:
- If f''(x) > 0, then the function is concave up (convex).
- If f''(x) < 0, then the function is concave down (concave).
Examples
Determine the concavity behavior of the function f(x) = x^2.
1. First derivative: f'(x) = 2x
2. Compute the second derivative: f''(x) = 2
Since f''(x) = 2 > 0, the function is concave up everywhere.
Determine the concavity behavior of the function f(x) = -x^3 + 3x.
1. First derivative: f'(x) = -3x^2 + 3
2. Compute the second derivative: f''(x) = -6x
3. Check: For x > 0 is f''(x) < 0 (concave down), for x < 0 is f''(x) > 0 (concave up).
Note
- ★The second derivative indicates the concavity of a function.
- ★If the second derivative is positive, the function is concave up; if it is negative, the function is concave down.
Exercises
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