Theory and Examples


In Exercises 51–54,


c. For what values of x, if any, is f' positive? Zero? Negative?


y = −x²

Verified step by step guidance

1. First, find the derivative of the function y = -x². The derivative, denoted as f', represents the rate of change of the function. Use the power rule for differentiation, which states that the derivative of x^n is n*x^(n-1).
2. Apply the power rule to y = -x². The derivative of -x² is f'(x) = -2x. This is because the derivative of x² is 2x, and the negative sign remains as a constant factor.
3. To determine where f' is positive, set the derivative -2x > 0 and solve for x. This inequality will tell you the range of x values where the function is increasing.
4. To find where f' is zero, set the derivative -2x = 0 and solve for x. This will give you the x value where the function has a horizontal tangent, indicating a potential maximum or minimum point.
5. To determine where f' is negative, set the derivative -2x < 0 and solve for x. This inequality will tell you the range of x values where the function is decreasing.

Key Concepts

Derivative

The derivative of a function, denoted as f', represents the rate of change of the function with respect to its variable. It provides information about the slope of the tangent line to the curve at any given point. For the function y = -x², the derivative helps determine where the function is increasing, decreasing, or has a horizontal tangent.

Critical Points

Critical points occur where the derivative of a function is zero or undefined. These points are important for identifying where a function changes from increasing to decreasing or vice versa. For y = -x², finding the critical points involves setting the derivative equal to zero and solving for x, which helps in analyzing the behavior of the function.

Sign of the Derivative

The sign of the derivative indicates whether a function is increasing or decreasing. If f' is positive, the function is increasing; if f' is negative, the function is decreasing; and if f' is zero, the function has a horizontal tangent. Analyzing the sign of the derivative for y = -x² helps determine the intervals where the function is increasing, decreasing, or constant.