Concavity, Slope, and Graphs

🔁 First, a quick refresher:

• A function is increasing when its slope is positive — it's going up as you move right.

• A function is decreasing when its slope is negative — it's going down as you move right.

• The rate at which a function increases or decreases depends on the second derivative, which tells us about the function's concavity.

🎯 What does it mean to:

1. Increase at a decreasing rate?

• The function is going up, but it's doing so more slowly as you move right.

• So the slope (first derivative) is positive but decreasing.

• This means the second derivative is negative.

• Therefore, the graph is concave down.

Example:
f(x)=xf(x) = \sqrt{x}f(x)=x​ or f(x)=ln⁡(x)f(x) = \ln(x)f(x)=ln(x) — these increase, but the rate slows down.

2. Decrease at an increasing rate?

• The function is going down, but it's doing so more steeply as you move right.

• So the slope is negative and becoming more negative.

• Again, the second derivative is negative.

• The graph is also concave down.

Example:
f(x)=−exf(x) = -e^xf(x)=−ex — it's decreasing, and the drop gets steeper.

3. Increase at an increasing rate?

• The function is going up, and going up faster and faster.

• The slope is positive and getting more positive.

• The second derivative is positive.

• The graph is concave up.

Example:
f(x)=exf(x) = e^xf(x)=ex or f(x)=x2f(x) = x^2f(x)=x2

4. Decrease at a decreasing rate?

• The function is going down, but it's slowing down its descent.

• The slope is negative, but getting less negative.

• So the second derivative is positive.

• The graph is concave up.

Example:
f(x)=−xf(x) = -\sqrt{x}f(x)=−x​ (on a domain where it's defined)

🧠 TL;DR — How it all connects: