Compute the derivative of the function using the familiar rules of calculus. If you have never taken calculus, you can use an online calculus calculator that computes the derivatives of functions. For instance, if you have the function f(x) = 5 + 4x - x^2, then the derivative is 4 - 2x.
Set the derivative equal to 0 and solve for x using basic algebra. For instance, the equation 4 - 2x = 0 has the solution x = 2.
Check that the solution for x is inside the closed interval that you are given and if so, proceed to the next step. Otherwise, skip to the next section. For instance, suppose the closed interval consists of all the numbers between 0 and 3, including 0 and 3. Since 2 is inside this interval, you must go to the next step.
Plug the number into the original function and record the value. For instance, when you plug 2 into the function f(x) = 5 + 4x - x^2, you get 9. This is because 5 + 4*2 - 2^2 equals 9.
Plug the number at the low end of the interval into the function and record the value. For example, suppose you are analyzing the function is f(x) = 5 + 4x - x^2 over the closed interval that consists of the numbers between 0 and 3, including 0 and 3. When you plug 0 in for x, you obtain 5, since 5 + 4*0 + 0^2 equals 5.
Plug the number at the high end of the interval into the function and record the value. When you plug 3 into the function, you obtain 8, since 5 + 4*3 - 3^2 equals 8.
Compare all of the numbers you obtained in all of the steps above and identify the highest and lowest numbers. These are the extrema. For instance, in this example you obtained the numbers 9, 5 and 8. This means that 9 is the maximum value of the function, and 5 is the lowest value of the function over the closed interval.