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Step 1
Find the derivative. For example, if we need the behavior of f(x) = x^2 + x, we use the power rule to find f '(x) = 2x + 1.
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Step 2
Find the critical numbers of the function. Determine where the derivative is undefined or equal to zero. In the example above, f '(x) is always defined (because it is a polynomial function). However, f '(x) does equal zero when x equals negative one-half as we see from the solution below.
f '(x) = 0
2x + 1 = 0
2x = -1
x = -1/2 -
Step 3
Determine the sign of f '(x) on either side (left/right) of every critical number and on either side of any discontinuity in the function f(x). In our example, we evaluate f '(x) for one value of x to the left of -1/2 and for one value to the right of -1/2. The choice is arbitrary. We see that f '(-1) is negative and that f '(0) is positive as below.
f '(-1) = 2(-1) + 1 = -2 + 1 = -1
f '(0) = 2(0) + 1 = 1 -
Step 4
State the solution. If f '(x) is negative to the left or right of a critical number of f(x) or to the left or right of a discontinuity of f(x), then f(x) decreases over that interval. Similarly, If f '(x) is positive to the left or right of a critical number of f(x) or to the left or right of a discontinuity of f(x), then f(x) increases over that interval. In our example, f '(x) was negative to the left of -1/2, so f(x) decreases over the interval on the x-axis from negative infinity to -1/2. Similarly, f '(x) was positive to the right of -1/2, which means f(x) increases over the interval on the x-axis from -1/2 to infinity.
