How to Determine the Concavity of a Function and Interpret the Results

Determine Concavity

To determine the concavity of a function, we need to look at the second derivative of the function

To determine the concavity of a function, we need to look at the second derivative of the function. The second derivative gives us information about the rate at which the slope of the function is changing.

Here are the steps to determine the concavity of a function:

1. Start by finding the first derivative of the function.
2. Then, find the second derivative of the function.
3. Set the second derivative equal to zero and solve.
4. Determine the intervals where the second derivative is positive or negative.
5. Interpret the concavity of the function based on the sign of the second derivative in those intervals.

Let’s go through an example to illustrate this process.

Example:
Given the function f(x) = x^3 – 6x^2 + 9x – 2, determine the concavity of the function.

Step 1: Find the first derivative:
f'(x) = 3x^2 – 12x + 9

Step 2: Find the second derivative:
f”(x) = 6x – 12

Step 3: Set the second derivative equal to zero and solve:
6x – 12 = 0
6x = 12
x = 2

Step 4: Determine the intervals where f”(x) is positive or negative. To do this, we choose test points within each interval and evaluate them in the second derivative.

Test point: x = 0
f”(0) = 6(0) – 12 = -12 (negative)

Test point: x = 3
f”(3) = 6(3) – 12 = 6 (positive)

Therefore, f”(x) is negative for x < 2 and positive for x > 2.

Step 5: Interpret the concavity based on the sign of the second derivative in those intervals. A positive second derivative indicates concavity upwards, while a negative second derivative indicates concavity downwards.

Since we found that f”(x) is negative for x < 2 and positive for x > 2, we can conclude that the function f(x) is concave downwards for x < 2 and concave upwards for x > 2.

In summary, the concavity of the function f(x) = x^3 – 6x^2 + 9x – 2 is concave downwards for x < 2 and concave upwards for x > 2.

More Answers: