Final answer:
To find the intervals of increasing and decreasing using the first derivative, find the critical points where the derivative equals zero or is undefined. Use the first derivative test to determine the intervals of increasing and decreasing.
Step-by-step explanation:
To find the intervals of increasing and decreasing using the first derivative, we need to find the critical points where the derivative equals zero or is undefined. First, let's find the first derivative of the given function:
y = -2x² + 4x + 3
y' = -4x + 4
Set y' equal to zero and solve for x to find the critical points:
-4x + 4 = 0
-4x = -4
x = 1
So the critical point is x = 1. Now we can use the first derivative test to determine the intervals of increasing and decreasing. Choose a value less than 1, like 0, and plug it into y'. If the result is positive, then the function is increasing. If the result is negative, then the function is decreasing.
y'(0) = -4(0) + 4 = 4
Since y'(0) is positive, the function is increasing in the interval (-∞, 1). Now choose a value greater than 1, like 2, and plug it into y'.
y'(2) = -4(2) + 4 = -4
Since y'(2) is negative, the function is decreasing in the interval (1, ∞).