Given the following functions below:
a. Y = -50 + 100x -5x²
b. Y = 10 - 16x + 3x²
Find the point at which the functions above maximized minimized. Also determine where point is maximization or minimization point.
To find the maximum or minimum points for each function, we need to find the derivative of the functions and set them equal to zero.
a. Y = -50 + 100x -5x²
Taking the derivative of Y with respect to x:
Y' = 100 - 10x
Setting Y' equal to zero and solving for x:
100 - 10x = 0
10x = 100
x = 10
To determine if this point is a maximum or minimum, we will use the second derivative test.
Taking the second derivative of Y:
Y'' = -10
Since the second derivative is negative, the point x = 10 is a maximum point for function a.
b. Y = 10 - 16x + 3x²
Taking the derivative of Y with respect to x:
Y' = -16 + 6x
Setting Y' equal to zero and solving for x:
-16 + 6x = 0
6x = 16
x ≈ 2.67
To determine if this point is a maximum or minimum, we will use the second derivative test.
Taking the second derivative of Y:
Y'' = 6
Since the second derivative is positive, the point x ≈ 2.67 is a minimum point for function b.