Topic 5.6 Notes – Determining Concavity of Functions over Their Domains

What Concavity Means

Think about how the slopes of tangent lines change as you move left to right.

• Concave up
  The graph looks like a cup opening upward.
  • Slopes are increasing as $x$ increases.
  • The derivative $f'(x)$ is increasing.
• Concave down
  The graph looks like a cap opening downward.
  • Slopes are decreasing as $x$ increases.
  • The derivative $f'(x)$ is decreasing.

Here’s the visual idea. In each panel, compare the curve $y = f(x)$ to its tangent line and notice how the slopes change as you move to the right.

Concave up and concave down, shown with tangent lines

Quick grounding reminder:

• $f'(x) > 0$ → function increasing
• $f'(x) < 0$ → function decreasing

Now we care about whether $f'(x)$ itself is increasing or decreasing. That’s where the second derivative comes in.

The Second Derivative

The second derivative is just the derivative of the derivative:

$$f''(x) = \frac{d}{dx}\big(f'(x)\big)$$

It measures how the slope is changing.

What the sign of $f''(x)$ tells you

• $f''(x) > 0$ → $f'$ increasing → concave up
• $f''(x) < 0$ → $f'$ decreasing → concave down
• $f''(x) = 0$ → possible change in concavity

On FRQs, graders want to see the logic spelled out. Not just “concave up,” but something like:

Since $f''(x) > 0$ on the interval, the graph of $f$ is concave up there.

That explicit connection earns the point.

Finding Intervals of Concavity

When you’re given a formula, the process is mechanical but easy to mess up if you rush.

1. Find $f'(x)$.
2. Find $f''(x)$.
3. Solve $f''(x) = 0$ or find where it’s undefined.
4. Use those values to split the number line.
5. Test the sign of $f''(x)$ in each interval.

Example:

Let $f(x) = x^3 - 4x$.

$$f'(x) = 3x^2 - 4$$ $$f''(x) = 6x$$

Set $6x = 0$ → $x = 0$.

Test intervals:

• Pick $x = -1$: $f''(-1) = -6$ → concave down
• Pick $x = 1$: $f''(1) = 6$ → concave up

Conclusion:

• Concave down on $(-\infty, 0)$
• Concave up on $(0, \infty)$

Always give answers in interval notation.

Points of Inflection

A point of inflection is where the function changes concavity.

Two conditions must be met:

1. $f''(c) = 0$ or undefined
2. $f''$ changes sign at $c$

Just solving $f''(x) = 0$ only gives a candidate.

From the example above:

• $f''(x) = 6x$ changes from negative to positive at $x = 0$.
  So there is an inflection point at
  $$(0, f(0)) = (0, 0)$$

Students lose points when they forget to check for the sign change. The AP will absolutely give you a case where $f''(x) = 0$ but concavity does not change.

Concavity from Graphs of $f'$ or $f''$

You won’t always get a formula. Sometimes you get a graph.

If you’re given the graph of $f'(x)$

Look at whether the derivative itself is rising or falling.

• Where $f'$ is increasing, $f$ is concave up.
• Where $f'$ is decreasing, $f$ is concave down.

You are watching the slope function rise or fall. In the graph above, $f'$ decreases until $x = 2$ and then increases, so $f$ changes from concave down to concave up at $x = 2$.

If you’re given the graph of $f''(x)$

• Above the x-axis → concave up
• Below the x-axis → concave down
• Crossing with sign change → inflection point

When working from a graph, always reference the derivative’s behavior, not how the original graph “looks.”