Understanding Concavity: Exploring the Relationship between Second Derivative and Graph Shape in Math

f”(x)>0

When the second derivative, f”(x), of a function is greater than 0, it tells us that the function is concave up

When the second derivative, f”(x), of a function is greater than 0, it tells us that the function is concave up. In other words, the graph of the function is shaped like a “U” or a cup that opens upward.

To better understand this concept, let’s break it down:

1. Definition of f”(x):
The second derivative, f”(x), of a function is the derivative of the first derivative, f'(x). It represents the rate at which the slope of the graph of the function is changing.

2. f”(x)>0:
When f”(x) is greater than 0, it means that the slope (or the rate of change) of the function’s tangent line is increasing. In other words, as you move from left to right along the graph of the function, the slope of the tangent line is becoming steeper.

3. Concave up:
This increasing slope behavior indicates that the graph of the function is bending upward or concave up. This concave shape resembles a “U” or a cup that opens upward.

4. Example:
Let’s consider a simple example: f(x) = x^2. Finding the first derivative, we get f'(x) = 2x. Taking the second derivative, we have f”(x) = 2. Since the second derivative, f”(x) = 2, is greater than 0, we conclude that the function f(x) = x^2 is concave up.

5. Graphical representation:
When we plot the graph of a function where f”(x) > 0, we observe that the curve is bending upward. The tangent lines to the curve have a positive slope that is getting steeper as we move along the x-axis.

Overall, when f”(x) > 0, it means that the function is concave up with a graph that is shaped like a “U” or a cup that opens upward. This concept is essential in understanding the behavior of functions and solving various mathematical problems.

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