Analyzing Concavity: How to Determine if a Function is Concave Up or Down

Determine Concavity

To determine the concavity of a function, we need to find the second derivative of the function and analyze its sign

To determine the concavity of a function, we need to find the second derivative of the function and analyze its sign.

Let’s say we have a function f(x). We can find the concavity using the following steps:

1. Find the first derivative of f(x) using differentiation rules.
2. Find the second derivative of f(x) by differentiating the first derivative obtained from step 1.
3. Analyze the sign of the second derivative.

If the second derivative is positive, the function is concave up. This means that the graph of the function curves upward, resembling the shape of a “U.”

If the second derivative is negative, the function is concave down. This means that the graph of the function curves downward, resembling the shape of an upside-down “U.”

If the second derivative equals zero, then the concavity is undefined at that point. This occurs where the function changes concavity.

To illustrate this, let’s consider an example using the function f(x) = 3x^2 – 6x + 2:

Step 1: Find the first derivative of f(x)
f'(x) = 6x – 6

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

Step 3: Analyze the sign of the second derivative
Since the second derivative f”(x) = 6 is a positive constant, the function f(x) = 3x^2 – 6x + 2 is concave up everywhere.

Alternatively, let’s consider another example using the function g(x) = x^3 – 6x^2 + 9x – 4:

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

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

Step 3: Analyze the sign of the second derivative
To find the critical point(s), we set g”(x) = 0 and solve for x:
6x – 12 = 0
x = 2

We can now analyze the sign of g”(x) in different intervals:
For x < 2: If we substitute x = 0 into g''(x), we get g''(0) = -12, which is negative. Thus, the graph of g(x) is concave down in this interval. For x > 2:
If we substitute x = 3 into g”(x), we get g”(3) = 6, which is positive. Hence, the graph of g(x) is concave up in this interval.

Therefore, the function g(x) = x^3 – 6x^2 + 9x – 4 is concave down when x < 2 and concave up when x > 2.

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