concavity and points of inflection examples grade 12

📚 Quick Study Guide

🔍 Concavity describes the direction a curve bends. A curve is concave up if it looks like a smile, and concave down if it looks like a frown. Mathematically, we look at the second derivative, $f''(x)$.
• 📈 If $f''(x) > 0$ on an interval, the function $f(x)$ is concave up on that interval.
• 📉 If $f''(x) < 0$ on an interval, the function $f(x)$ is concave down on that interval.
• 📍 A point of inflection is a point on the curve where the concavity changes. This occurs where $f''(x) = 0$ or $f''(x)$ is undefined, and the concavity changes sign around that point.
• 📝 To find points of inflection: First, find the second derivative, $f''(x)$. Set $f''(x) = 0$ and solve for $x$. Check the sign of $f''(x)$ on either side of these $x$ values to confirm a change in concavity. Finally, plug these $x$ values back into the original function $f(x)$ to find the corresponding $y$ values.
• 📐 The second derivative test can also be used to find local extrema: If $f'(c) = 0$ and $f''(c) > 0$, then $f(c)$ is a local minimum. If $f'(c) = 0$ and $f''(c) < 0$, then $f(c)$ is a local maximum.

🧪 Practice Quiz

1. What does it mean if $f''(x) > 0$ on an interval?
1. $f(x)$ is concave down.
2. $f(x)$ is concave up.
3. $f(x)$ has a point of inflection.


2. At a point of inflection:
1. $f'(x) = 0$
2. The concavity of $f(x)$ changes.
3. $f''(x) > 0$


3. If $f'(c) = 0$ and $f''(c) < 0$, then $f(c)$ is:
1. A local minimum.
2. A local maximum.
3. A point of inflection.


4. For $f(x) = x^3$, where is the possible point of inflection?
1. $x = 0$
2. $x = 1$
3. $x = -1$


5. Given $f(x) = x^4$, is there a point of inflection at $x = 0$?
1. Yes, because $f''(0) = 0$.
2. No, because the concavity doesn't change.
3. Yes, because $f(0) = 0$.


6. The concavity of the curve $y = x^2 + x$ is:
1. Always concave up.
2. Always concave down.
3. Changes at $x = 0$.


7. Find the interval where the function $f(x) = x^3 - 6x^2 + 5x$ is concave up.
1. $(-\infty, 2)$
2. $(2, \infty)$
3. $(-\infty, \infty)$