Find the Maximum and Minimum of a Function Algebraically

The point at which a graph changes direction is called a maximum or a minimum.

Steps

1. Take the first derivative of the function.
2. Find the critical numbers of the function. You can find the critical numbers by letting the derivative equal zero and solving and then letting the derivative be undefined and then solving. There may be multiple solutions.

Using A Number Line Graph

1. Plot these solutions on a number line graph. Draw a number line graph with tick marks at your solutions. Label the top of these ticks with 0 or und, representing the value of the derivative at these values. Also label the number line as the derivative of your function.
2. Find whether each interval between critical numbers is positive or negative. Plug a value that lies in each interval between critical numbers into your derivative and see whether this value is positive or negative. The sign on this number is the sign of the interval.
3. Check if the derivative changes signs at the critical numbers. See if the intervals on either side of each critical number are different signs. If the derivative changes from positive to negative, the function has a relative maximum, and if the derivative changes from negative to positive, the function has a relative minimum.
4. For the values found to be relative extrema, substitute x back into the function to find the equivalent y value. This is the value of the relative maximum/minimum.

Using the Second Derivative Test

1. Take the second derivative of the function.
2. Substitute each critical number into the second derivative and find its value.
   • If the second derivative is positive, that point is a minimum - the graph turns upwards at that point.
   • If the second derivative is negative, that point is a maximum - the graph turns downwards at that point.
   • If the second derivative is zero, that point is a point of inflection and is not a relative extremum - the graph levels out and then continues in the same direction.
3. For the values found to be relative extrema, substitute x back into the function to find the equivalent y value. This is the value of the relative maximum/minimum.

See also

• Use the Slope Intercept Form (in Algebra)
• Start Working with Continued Fractions
• Convert a Quadratic Formula to Roots Form by Completing the Square
• Factor Second Degree Polynomials (Quadratic Equations)
• Understand Linear Equations
• Understand Slope (in Algebra)
• Prove That the Square Root of Two Is Irrational
• Factor a Cubic Polynomial
• Calculate Frequency
• Find the Maximum or Minimum Value of a Quadratic Function Easily

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