Intervals of Increase and Decrease

Analyzing the Function f(x) = 4x^2 - 2

To determine where a function is increasing or decreasing, we analyze its derivative and the behavior of its graph.

• Quadratic functions of the form f(x) = ax^2 + bx + c are parabolas. If a > 0, the parabola opens upward; if a < 0, it opens downward.

• The function increases where its derivative is positive and decreases where its derivative is negative.

Step 1: Find the Derivative

• The derivative of f(x) = 4x^2 - 2 is:

Step 2: Find Critical Points

• Set the derivative equal to zero to find critical points:

Step 3: Test Intervals

• For x < 0: is negative, so the function is decreasing.

• For x > 0: is positive, so the function is increasing.

Step 4: State the Interval Where f is Increasing

• The function f(x) = 4x^2 - 2 is increasing on the interval:

Example

• For x = 1: , so the function is increasing at x = 1.

• For x = -1: , so the function is decreasing at x = -1.

Summary Table: