Given the function below:
a. Y = -50 + 100x - 5x²
b. Y = 10 - 16x +3x²
Find the point at which the functions above is maximized or minimized. Also determine whether the point is maximization or minimization point.
To find the max/min point of each function, we need to find the derivative of each function with respect to x, set it equal to 0, and solve for x.
a. Y = -50 + 100x - 5x²
The derivative of Y with respect to x is:
Y' = 100 - 10x
Setting Y' equal to 0:
100 - 10x = 0
10x = 100
x = 10
To determine if this point is a maximum or minimum, we can look at the second derivative of Y:
Y'' = -10
Since the second derivative is negative, the point x = 10 is a maximum point for function Y.
b. Y = 10 - 16x + 3x²
The derivative of Y with respect to x is:
Y' = -16 + 6x
Setting Y' equal to 0:
-16 + 6x = 0
6x = 16
x = 16/6
x = 8/3
To determine if this point is a maximum or minimum, we can look at the second derivative of Y:
Y'' = 6
Since the second derivative is positive, the point x = 8/3 is a minimum point for function Y.