Question

On the right, you have a graph of f(x). Answer the queens.

A. At what x values does f(x) have a horizontal tangent? Explain your solution.
B. On what interval of f(x) is f(x) > 0? Explain your solution.
C. At what x values does f(x) have a relative extrema? Explain your solution.

On the right, you have a graph of g'(x). Answer the following questions.
A. At what x values does g(x) have a horizontal tangent? Explain your solution.
B. On what interval of g'(x) is g(x) > 0?

Explain your solution.

Answer 1

f(x) has a horizontal tangent at every point since the graph is a horizontal line. f(x) > 0 for all x values between 0 and 20. The graph of g'(x) intersects the x-axis at x values where g(x) has a horizontal tangent, and g(x) > 0 for the interval between those x values.

Step-by-step explanation:

A. To determine the x values where f(x) has a horizontal tangent, we need to find where the slope of the function is zero. Since the graph of f(x) is a horizontal line, the slope is always zero. Therefore, f(x) has a horizontal tangent at every point.

B. Since f(x) is a horizontal line, it is equal to a constant value. If the constant value is greater than 0, then f(x) > 0. In this case, since the graph of f(x) is always above the x-axis, f(x) > 0 for all x values between 0 and 20, inclusive.

C. Since the graph of f(x) is a horizontal line, it does not have any relative extrema. The function is constant and does not have any points of maximum or minimum.

D. To determine the x values where g(x) has a horizontal tangent, we need to find where the slope of the function is zero. In the graph of g'(x), the x values where the graph intersects the x-axis are the values where g(x) has a horizontal tangent.

E. To determine the interval of g'(x) where g(x) > 0, we need to find where the graph of g'(x) is above the x-axis. In this case, the graph of g'(x) is above the x-axis for the interval between the two x values where it intersects the x-axis.