Concavity of Functions (College Board AP® Calculus AB): Revision Note

Concavity of functions

What is concavity?

• Concavity is the way in which a curve bends and is related to the second derivative of a function

• A curve is:

  • Concave up if $f^{\text{'}}\left(x\right)$ is increasing in this interval

    • $f^{\text{'}\text{'}}\left(x\right)\ge 0$ for all values of$x$ in an interval

  • Concave down if $f^{\text{'}}\left(x\right)$ is decreasing in this interval

    • $f^{\text{'}\text{'}}\left(x\right)\le 0$ for all values of$x$ in an interval

How do I find where a function is concave up or concave down using the second derivative?

• The second derivative of a function, $f\text{'}\text{'}\left(x\right)$, describes the rate of change of the first derivative, $f\text{'}\left(x\right)$

  • If the rate of change is positive, the first derivative is increasing

  • If the rate of change is negative, the first derivative is decreasing

• This means you can determine if a function is concave up or down at a point using the second derivative

  • If $f\text{'}\text{'}\left(a\right)>0$ then $f$ is concave up at $x=a$

  • If $f\text{'}\text{'}\left(a\right)<0$ then $f$ is concave down at $x=a$

• You can also find an interval where a function is concave up or down

  • To find where the function is concave up,

    • Solve the inequality $f\text{'}\text{'}\left(x\right)>0$

  • To find where the function is concave down,

    • Solve the inequality $f\text{'}\text{'}\left(x\right)<0$

What happens if the second derivative is zero?

• If $f\text{'}\text{'}\left(a\right)=0$ then the function could be concave up, concave down, or neither at these points

• You determine what is happening at this point by looking at the sign of the second derivative on either side of this point

  • If the second derivative is positive on either side of the point, then $f$ is concave up at $x=a$

    • Consider $f\left(x\right)=x^4$

    • $f\text{'}\text{'}\left(0\right)=0$

    • $f\text{'}\text{'}\left(-0.01\right)>0$ and $f\text{'}\text{'}\left(0.01\right)>0$

    • Therefore, $f$ is concave up at (0, 0)

  • If the second derivative is negative on either side of the point, then $f$ is concave down at $x=a$

    • Consider $f\left(x\right)=-x^4$

    • $f\text{'}\text{'}\left(0\right)=0$

    • $f\text{'}\text{'}\left(-0.01\right)<0$ and $f\text{'}\text{'}\left(0.01\right)<0$

    • Therefore, $f$ is concave down at (0, 0)

  • If the sign of the second derivative changes on either side of the point, then $f$ is neither concave up nor concave down

    • Consider $f\left(x\right)=x^3$

    • $f\text{'}\text{'}\left(0\right)=0$

    • $f\text{'}\text{'}\left(-0.01\right)<0$ and $f\text{'}\text{'}\left(0.01\right)>0$

    • Therefore, $f$ is neither concave up nor concave down at (0, 0)

    • It is called a point of inflection

How do I find points of inflection?

• Find the points where the second derivative is zero

  • The value of the first derivative could be any value

• Find the sign of the second derivative around the points to find whether the concavity changes

  • Check whether it goes from concave up to concave down or vice versa

  • If it does, then the point is a point of inflection