Increasing & Decreasing Functions (College Board AP® Calculus AB): Revision Note

Increasing & decreasing functions

What are increasing and decreasing functions?

• A non-constant function $f$ is increasing on the interval $\left[a,b\right]$ if:

  • $f\left(x_1\right)\le f\left(x_2\right)$ whenever $x_1<x_2$

  • If $f\left(x_1\right)<f\left(x_2\right)$ whenever $x_1<x_2$, then the function is strictly-increasing

• A non-constant function $f$ is decreasing on the interval $\left[a,b\right]$ if:

  • $f\left(x_1\right)\ge f\left(x_2\right)$ whenever $x_1<x_2$

  • If $f\left(x_1\right)>f\left(x_2\right)$ whenever $x_1<x_2$, then the function is strictly-decreasing

• A constant function is neither increasing nor decreasing

How do I find where a function is increasing and decreasing using the first derivative?

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

  • If the rate of change is positive, the function is increasing

  • If the rate of change is negative, the function is decreasing

• This means you can determine if a function is increasing or decreasing at a point using the first derivative

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

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

• You can also find an interval where a function is increasing or decreasing

  • To find where the function is increasing,

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

  • To find where the function is decreasing,

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

What happens if the first derivative is zero?

• If $f\text{'}\left(a\right)=0$ then there is a critical point at $x=a$

  • The function could be increasing, decreasing, or neither at these points

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

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

    • Consider $f\left(x\right)=x^3$

    • $f\text{'}\left(0\right)=0$

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

    • Therefore, $f$ is increasing at (0, 0)

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

    • Consider $f\left(x\right)=-x^3$

    • $f\text{'}\left(0\right)=0$

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

    • Therefore, $f$ is decreasing at (0, 0)

  • If the sign of the derivative changes on either side of the point, then $f$ is neither increasing nor decreasing

    • Consider $f\left(x\right)=x^2$

    • $f\text{'}\left(0\right)=0$

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

    • Therefore, $f$ is neither increasing nor decreasing at (0, 0)

  • If the derivative is also zero on either side of the point, then $f$ is constant on an interval

    • Therefore, $f$ is neither increasing nor decreasing

How can I identify intervals where the function is increasing or decreasing by using the graph of the derivative?

• Sketching a graph of both $f\left(x\right)$ and $f\text{'}\left(x\right)$ can help to identify where a function will be increasing or decreasing

  • On the graph of $f\left(x\right)$,

    • An upward slope from left to right is where the function is increasing

    • A downward slope from left to right is where the function is decreasing

  • On the graph of $f\text{'}\left(x\right)$,

    • The portion of the graph above the $x$-axis is where the function is increasing

    • The portion of the graph below the $x$-axis is where the function is decreasing

• The diagram below shows a cubic and its derivative, a quadratic, plotted on the same graph

  • Between the critical points at $a$ and $b$, the cubic is decreasing

  • Therefore, the graph of the derivative is below the $x$-axis between $a$ and $b$