find the intervals of y=2x^2 - 19x+35

Answer 1

To find the intervals of the function y = 2x^2 - 19x + 35, we need to determine where the function is increasing or decreasing.

Step 1: Find the critical points by setting the derivative equal to zero and solving for x.
The derivative of y = 2x^2 - 19x + 35 is dy/dx = 4x - 19.
Setting this equal to zero, we have 4x - 19 = 0.
Solving for x, we get x = 19/4 = 4.75.

Step 2: Determine the intervals using the critical point.
Since the critical point is x = 4.75, we can use a number line to determine the intervals.

Choosing a number less than 4.75, such as x = 0, we can evaluate the sign of the derivative.
dy/dx = 4(0) - 19 = -19. Since this is negative, the function is decreasing in the interval (-∞, 4.75).

Choosing a number greater than 4.75, such as x = 5, we can evaluate the sign of the derivative.
dy/dx = 4(5) - 19 = 1. Since this is positive, the function is increasing in the interval (4.75, ∞).

Therefore, the intervals of the function y = 2x^2 - 19x + 35 are (-∞, 4.75) and (4.75, ∞).