Given the following functions:
a. Y = 2 - x²
b. Y = 2x² - 8x + 6
c. Y = 2x³ + 3x² - 12x + 8
Find the point at which the functions above maximized or minimized. Also determine whether the point is maximization or minimization point.
To find the points where the functions are maximized or minimized, we need to find the derivative of each function and set it equal to zero to solve for critical points.
a. Y = 2 - x²
Derivative:
Y' = -2x
Setting Y' equal to zero:
-2x = 0
x = 0
Plug x = 0 back into the original function:
Y = 2 - 0²
Y = 2
So the point at which function a is maximized or minimized is (0, 2). To determine if it is a maximum or minimum, we can check the second derivative:
Y'' = -2
Since the second derivative is negative, the point (0, 2) is a maximum point for function a.
b. Y = 2x² - 8x + 6
Derivative:
Y' = 4x - 8
Setting Y' equal to zero:
4x - 8 = 0
4x = 8
x = 2
Plug x = 2 back into the original function:
Y = 2(2)² - 8(2) + 6
Y = 8 - 16 + 6
Y = -2
So the point at which function b is maximized or minimized is (2, -2). To determine if it is a maximum or minimum, we can check the second derivative:
Y'' = 4
Since the second derivative is positive, the point (2, -2) is a minimum point for function b.
c. Y = 2x³ + 3x² - 12x + 8
Derivative:
Y' = 6x² + 6x - 12
Setting Y' equal to zero:
6x² + 6x - 12 = 0
Divide by 6:
x² + x - 2 = 0
(x + 2)(x - 1) = 0
x = -2 or x = 1
Plug x = -2 back into the original function:
Y = 2(-2)³ + 3(-2)² - 12(-2) + 8
Y = -16 + 12 + 24 + 8
Y = 28
Plug x = 1 back into the original function:
Y = 2(1)³ + 3(1)² - 12(1) + 8
Y = 2 + 3 - 12 + 8
Y = 1
So the points at which function c is maximized or minimized are (-2, 28) and (1, 1). To determine if they are maximum or minimum points, we can check the second derivative:
Y'' = 12x + 6
For x = -2:
Y'' = 12(-2) + 6
Y'' = -24 + 6
Y'' = -18 (negative)
For x = 1:
Y'' = 12(1) + 6
Y'' = 12 + 6
Y'' = 18 (positive)
Therefore, the point (-2, 28) is a maximum point and the point (1, 1) is a minimum point for function c.