Concavity for MATH 139
Exam Relevance for MATH 139
Concavity and inflection points appear on every MATH 139 exam. Use f''(x) to determine where the function curves up or down.
What is Concavity?
Concavity describes the "curvature" of a function — whether it curves upward like a bowl or downward like a hill.
The Big Idea: The second derivative tells you which way a function is curving.
Concave Up vs Concave Down
Concave Up (Smiley Face 😊)
A function is concave up when it curves upward — like a bowl that holds water.
- The slope is increasing (getting steeper upward or less steep downward)
- $f''(x) > 0$
- Tangent lines lie below the curve
Notice how the parabola $f(x) = x^2$ opens upward. Any tangent line you draw will be below the curve (except at the point of tangency).
Concave Down (Frowny Face ☹️)
A function is concave down when it curves downward — like an upside-down bowl.
- The slope is decreasing (getting less steep upward or steeper downward)
- $f''(x) < 0$
- Tangent lines lie above the curve
The parabola $f(x) = -x^2$ opens downward. Tangent lines lie above the curve.
The Second Derivative Test for Concavity
| If... | Then the function is... |
|---|---|
| $f''(x) > 0$ | Concave UP on that interval |
| $f''(x) < 0$ | Concave DOWN on that interval |
Why Does This Work?
- $f''(x) = $ rate of change of $f'(x) = $ rate of change of the slope
- If $f''(x) > 0$, the slope is increasing → concave up
- If $f''(x) < 0$, the slope is decreasing → concave down
Inflection Points
An inflection point is where concavity changes — the function switches from concave up to concave down (or vice versa).
How to Find Inflection Points
- Find $f''(x)$
- Set $f''(x) = 0$ and solve (also check where $f''(x)$ is undefined)
- Test points on either side to confirm concavity actually changes
- Find the $y$-coordinate: plug the $x$-value back into $f(x)$
⚠️ Important: $f''(x) = 0$ is necessary but NOT sufficient for an inflection point. You must verify the concavity actually changes!
Find the intervals where $f(x) = x^3 - 6x^2 + 9x + 1$ is concave up and concave down.
Step 1: Find the second derivative
$f(x) = x^3 - 6x^2 + 9x + 1$
$f'(x) = 3x^2 - 12x + 9$
$f''(x) = 6x - 12$
Step 2: Find where $f''(x) = 0$
$6x - 12 = 0$
$x = 2$
Step 3: Test intervals
| Interval | Test point | $f''(x) = 6x - 12$ | Concavity |
|---|---|---|---|
| $(-\infty, 2)$ | $x = 0$ | $6(0) - 12 = -12 < 0$ | Down |
| $(2, \infty)$ | $x = 3$ | $6(3) - 12 = 6 > 0$ | Up |
$$\boxed{\text{Concave up: } (2, \infty) \quad \text{Concave down: } (-\infty, 2)}$$
Find the inflection point(s) of $f(x) = x^4 - 4x^3 + 6$.
Step 1: Find the second derivative
$f'(x) = 4x^3 - 12x^2$
$f''(x) = 12x^2 - 24x = 12x(x - 2)$
Step 2: Find where $f''(x) = 0$
$12x(x - 2) = 0$
$x = 0$ or $x = 2$
Step 3: Test concavity around each point
| Interval | Test point | $f''(x) = 12x(x-2)$ | Concavity |
|---|---|---|---|
| $(-\infty, 0)$ | $x = -1$ | $12(-1)(-3) = 36 > 0$ | Up |
| $(0, 2)$ | $x = 1$ | $12(1)(-1) = -12 < 0$ | Down |
| $(2, \infty)$ | $x = 3$ | $12(3)(1) = 36 > 0$ | Up |
Concavity changes at both $x = 0$ and $x = 2$ ✓
Step 4: Find the $y$-coordinates
$f(0) = 0 - 0 + 6 = 6$
$f(2) = 16 - 32 + 6 = -10$
$$\boxed{\text{Inflection points: } (0, 6) \text{ and } (2, -10)}$$
Show that $f(x) = x^4$ has no inflection point at $x = 0$.
Step 1: Find the second derivative
$f'(x) = 4x^3$
$f''(x) = 12x^2$
Step 2: Find where $f''(x) = 0$
$12x^2 = 0 \implies x = 0$
Step 3: Test concavity
| Interval | Test point | $f''(x) = 12x^2$ | Concavity |
|---|---|---|---|
| $(-\infty, 0)$ | $x = -1$ | $12(1) = 12 > 0$ | Up |
| $(0, \infty)$ | $x = 1$ | $12(1) = 12 > 0$ | Up |
Concavity does NOT change! Both sides are concave up.
$$\boxed{\text{No inflection point at } x = 0}$$
Find the intervals of concavity for $f(x) = \frac{x}{x^2 + 1}$.
Step 1: Find the first derivative (quotient rule)
$$f'(x) = \frac{(1)(x^2+1) - (x)(2x)}{(x^2+1)^2} = \frac{x^2 + 1 - 2x^2}{(x^2+1)^2} = \frac{1 - x^2}{(x^2+1)^2}$$
Step 2: Find the second derivative (quotient rule again)
Let $u = 1 - x^2$ and $v = (x^2+1)^2$
$u' = -2x$
$v' = 2(x^2+1)(2x) = 4x(x^2+1)$
$$f''(x) = \frac{-2x(x^2+1)^2 - (1-x^2) \cdot 4x(x^2+1)}{(x^2+1)^4}$$
Factor out $2x(x^2+1)$ from numerator:
$$= \frac{2x(x^2+1)[-(x^2+1) - 2(1-x^2)]}{(x^2+1)^4} = \frac{2x[-x^2-1-2+2x^2]}{(x^2+1)^3} = \frac{2x(x^2 - 3)}{(x^2+1)^3}$$
Step 3: Find where $f''(x) = 0$
$2x(x^2 - 3) = 0$
$x = 0$ or $x = \pm\sqrt{3}$
Step 4: Test intervals (denominator is always positive)
| Interval | Sign of $2x$ | Sign of $(x^2-3)$ | $f''(x)$ | Concavity |
|---|---|---|---|---|
| $(-\infty, -\sqrt{3})$ | − | + | − | Down |
| $(-\sqrt{3}, 0)$ | − | − | + | Up |
| $(0, \sqrt{3})$ | + | − | − | Down |
| $(\sqrt{3}, \infty)$ | + | + | + | Up |
$$\boxed{\text{Concave up: } (-\sqrt{3}, 0) \cup (\sqrt{3}, \infty)}$$ $$\boxed{\text{Concave down: } (-\infty, -\sqrt{3}) \cup (0, \sqrt{3})}$$
Tips for Concavity Problems
- Second derivative = concavity — always find $f''(x)$ first
- Make a sign chart — organized tables help track sign changes
- Check both conditions for inflection points: $f''(x) = 0$ AND concavity changes
- Factor $f''(x)$ when possible — makes sign analysis easier
- Don't forget undefined points — if $f''(x)$ is undefined, check there too
Common Mistakes and Misunderstandings
❌ Mistake: Assuming $f''(x) = 0$ always gives an inflection point
Wrong: "$f''(0) = 0$ for $f(x) = x^4$, so $(0, 0)$ is an inflection point."
Why it's wrong: You must check that concavity actually changes. For $x^4$, it's concave up on both sides of $x = 0$.
Correct: Test intervals on both sides. If concavity doesn't change, it's NOT an inflection point.
❌ Mistake: Confusing increasing/decreasing with concave up/down
Wrong: "The function is going up, so it's concave up."
Why it's wrong: A function can be increasing while concave down (like the left half of $-x^2 + 10$ near the peak).
Correct: Increasing/decreasing is about $f'(x)$. Concavity is about $f''(x)$.
❌ Mistake: Forgetting to find the $y$-coordinate of inflection points
Wrong: "The inflection point is at $x = 2$."
Why it's wrong: An inflection point is a point on the curve, so you need both coordinates.
Correct: Plug $x = 2$ back into $f(x)$ to get the $y$-value. Report as $(2, f(2))$.
Second Derivative Test for Concavity
The sign of the second derivative tells you the concavity. Positive means the curve opens upward (like a bowl); negative means it opens downward (like a hill).
Variables:
- $f''(x)$:
- The second derivative of f
- $\text{Concave Up}$:
- Curve opens upward, tangent lines below curve
- $\text{Concave Down}$:
- Curve opens downward, tangent lines above curve
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