Understanding the Relationship Between a Concave Up Function and a Positive Second Derivative

If f(x) is concave up, then f”(x) is?

If a function f(x) is concave up, then f”(x) is positive

If a function f(x) is concave up, then f”(x) is positive.

To understand why this is true, let’s define concavity and examine the behavior of the second derivative.

A function f(x) is concave up on an interval if, for any two points within that interval, the graph of the function lies above the line segment connecting those two points. Geometrically, this means that the function is curving upward or “opening up” on that interval.

Now, let’s consider the second derivative, denoted as f”(x). The second derivative measures the rate of change of the first derivative. In simpler terms, it tells us how the slope of the tangent line to the graph of f(x) changes as x varies.

When f”(x) is positive, it means that the slope of the tangent line is increasing as x varies. In other words, the graph of f(x) is “bending” more and more upward, indicating a concave up behavior.

Here’s an intuitive way to understand the relationship:

Imagine a function f(x) that is concave up.
– Suppose you take two points on the graph, (x1, f(x1)) and (x2, f(x2)).
– The line connecting these two points will be a secant line.
– If f(x) is concave up, then the graph of the function lies above this secant line.
– As you move the two points closer together, the secant line becomes the tangent line at a particular point (let’s say x0), and the area between the graph and the tangent line becomes smaller.
– This indicates that the tangent line’s slope is increasing, which corresponds to a positive second derivative.

In summary, if a function f(x) is concave up, it means that f”(x) is positive, indicating that the slope of the tangent line is increasing as x changes.

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