Brief Calculus.

Question

Sketch f (x) = 2x²-7

a) Find the x-values where there are horizontal tangent lines

b) Find the intervals of x where f(x) is increasing

c) Find the intervals of x where f(x) is decreasing.

Answer 1

To find the answers, you must take the derivative of f(x).

If f(x) = 2x² - 7, then f '(x) = 4x (using the power rule).

The derivative is the slope of the tangent line at any value of "x". To find the x-value where the slope of the tangent line is zero (i.e., a horizontal line), then set f '(x) equal to zero and solve for "x"

4x = 0

x = 0

So the x-value where there is a horizontal tangent line is at x = 0

To find the locations where the function is increasing, you want to know what the x-values are that result in positive tangent line slopes so you want 4x > 0. That occurs when x > 0, so on the interval (0, ∞)

To find the locations where the function is decreasing, you want to know what the x-values are that result in negative tangent line slopes so you want 4x < 0. That occurs when x < 0, so on the interval (-∞, 0)