Find Local Extrema Using the First Derivative — Practice Questions - dummies

Find Local Extrema Using the First Derivative — Practice Questions

By Mark Ryan

You can use a derivative to locate the top of a “hill” and the bottom of a “valley,” called local extrema, on just about any function. You simply set the derivative of the function equal to zero and solve for x.

A derivative is useful because it tells you the slope of a curve. This means that any problem involving anything about the slope or steepness of a curve is a derivative question.

Practice questions

  1. Use the first derivative to find the local extrema of f(x) = 6x2/3 – 4x + 1.

  2. Find the local extrema of

    A function with a trigonometric identity

    in the interval

    A mathematical interval, zero to two pi.

    with the first derivative test.

Answers and explanations

  1. Using the first derivative of f(x) = 6x2/3 – 4x + 1, the local min is at (0, 1), and the local max is at (1, 3).

    To find these local extrema, you start by finding the first derivative using the power rule.

    finding the first derivative using the power rule.

    Now you find the critical numbers of f. First, you set the derivative equal to zero and solve:

    Set the derivative equal to zero and solve

    Then, you need to determine the x values where the derivative is undefined.

    Determine the x values where the derivative is undefined.

    Because the denominator is not allowed to equal zero,

    The first derivative of a function.

    is undefined at x = 0. Thus the critical numbers of f are 0 and 1. You can plot these critical numbers on a number line if it helps.

    Plug a number from each of the three regions into the derivative.

    Plug number from each of the three regions to find if that section of the graph is positive or negative.

    Note how the numbers that were chosen for the first and third computations made the math easy. With the second computation, you can save a little time and skip the final calculation because all you care about is whether the result is positive or negative (this assumes that you know that the cube root of 2 is more than 1 — you’d better!).

    Draw your sign graph.

    A sign graph for a function.

    Determine whether there’s a local min or max or neither at each critical number.

    f goes down to where x = 0 and then up, so there’s a local min at x = 0, and f goes up to where x = 1 and then down, so there’s a local max at x = 1.

    Figure the y value of the two local extrema.

    Figuring the y value of the two local extrema

    Thus, there’s a local min at (0, 1) and a local max at (1, 3). Check this answer by looking at a graph of f on your graphing calculator.

  2. Using the first derivative test, for

    A trigonometric function.

    in the interval

    The interval zero to two times Pi

    the local max is at

    Local maximum for a function.

    and the local min is at

    Local minimum for a function.

    And here’s how the graph looks.

    A graph for a function with an interval of zero to two.

    Find the first derivative.

    Finding the first derivative for a function.

    Find the critical numbers of h.

    Set the derivative equal to zero and solve:

    Setting the first derivative of a function to zero and solving it.

    Determine the x values where the derivative is undefined.

    The derivative isn’t undefined anywhere, so the critical numbers of h are

    Critical numbers of a function.

    Test numbers from each region on your number line.

    Testing numbers from each region on the number line

    Draw a sign graph.

    Drawing a sign graph for a function.

    Decide whether there’s a local min, max, or neither at each of the two critical numbers.

    Going from left to right along the function, you go up until

    The local maximum for a function

    and then down, so there’s a local max at

    Local maximum x-point for a function.

    It’s vice versa for

    The local minimum point for a function.

    so there’s a local min there.

    Compute the y values of these two extrema.

    Computing the y values for the local extrema.

    So you have a max at

    A graph's local maximum

    and a min at

    The local minimum point of a graph.