Concavity for MATH 139
Exam Relevance for MATH 139
Use second derivative: f double prime > 0 means concave up, < 0 means concave down. Find inflection points.
What is Concavity?
Concavity describes the "curvature" of a function â 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 đ)
A function is concave up when it curves upward â 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 âšī¸)
A function is concave down when it curves downward â 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 â concave up
- If $f''(x) < 0$, the slope is decreasing â concave down
Inflection Points
An inflection point is where concavity changes â 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 $f(x)$
â ī¸ 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 $x = 2$ â
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 |
|---|---|---|---|---|
| $(-\infty, -\sqrt{3})$ | â | + | â | Down |
| $(-\sqrt{3}, 0)$ | â | â | + | Up |
| $(0, \sqrt{3})$ | + | â | â | 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 â always find $f''(x)$ first
- Make a sign chart â organized tables help track sign changes
- Check both conditions for inflection points: $f''(x) = 0$ AND concavity changes
- Factor $f''(x)$ when possible â makes sign analysis easier
- Don't forget undefined points â if $f''(x)$ is undefined, check there too
Common Mistakes and Misunderstandings
â 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.
â 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 $f''(x)$.
â 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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