Find the absolute maximum and absolute minimum values of f on the given interval.
f(x) = 2x^3 − 3x^2 − 72x + 7 , [−4, 5]
f'(x) = 6x^2 -6x -72
f'(0) = 6(x^2 -x-12)
0 = 6(x+ 3) (x-4)
x = -3, 4
f(-4) = 2(-4)^3 -3(-4)^2 -72(-4) +7= 139
f(-3) = 2(-3)^3 -3(-3)^2 -72(-3) +7=132
f(4) = 2(4)^3 -3(4)^2 -72(4) + 7= -145
f(5) = 2(5)^3 -3(5)^2 -72(5)+7= -178
f ' (x) = 6x^2 - 6x - 72
= 0 for a local max/min
x^2 - x - 12 = 0
(x-4)(x+3) = 0
x = 4, or x = -3
f(4) = 2(64) - 3(16) - 72(4) + 7 = -201
f(-3) = 2(-27) - 3(9) - 72(-3)+7 = 115
end-values:
f(-4) = 2(-64) - 2(16)( - 72(-4) + 7 = 135
f(5) = 2(125) - 3(25) - 72(5) + 7 = -178
so what do you think ?
looks like we both messed up in the f(x) calculations
should be
f(4) = 2(64) - 3(16) - 72(4) + 7 = -201
f(-3) = 2(-27) - 3(9) - 72(-3)+7 = 142
end-values:
f(-4) = 2(-64) - 3(16)( - 72(-4) + 7 = 119
f(5) = 2(125) - 3(25) - 72(5) + 7 = -178
To find the absolute maximum and minimum values of a function on a given interval, we can follow these steps:
1. Find the critical points of the function within the given interval by taking its derivative and solving for when it equals zero or is undefined.
2. Evaluate the function at the critical points and at the endpoints of the interval.
3. Compare the function values to determine the absolute maximum and minimum values.
Let's apply these steps to the function f(x) = 2x^3 − 3x^2 − 72x + 7 on the interval [-4, 5]:
Step 1: Find the critical points:
Take the derivative of f(x) to find its critical points:
f'(x) = 6x^2 - 6x - 72
Set f'(x) equal to zero and solve for x:
6x^2 - 6x - 72 = 0
Simplify the equation:
x^2 - x - 12 = 0
Factor the equation:
(x - 4)(x + 3) = 0
Solve for x:
x = 4 or x = -3
Step 2: Evaluate the function at the critical points and endpoints:
Evaluate f(x) at the critical points (x = 4 and x = -3):
f(4) = 2(4)^3 − 3(4)^2 − 72(4) + 7 = 98
f(-3) = 2(-3)^3 − 3(-3)^2 − 72(-3) + 7 = -98
Evaluate f(x) at the endpoints of the interval [-4, 5]:
f(-4) = 2(-4)^3 − 3(-4)^2 − 72(-4) + 7 = -271
f(5) = 2(5)^3 − 3(5)^2 − 72(5) + 7 = 78
Step 3: Compare the function values:
The function values on the interval are:
f(-4) = -271
f(-3) = -98
f(4) = 98
f(5) = 78
The absolute maximum value is 98 at x = 4 and the absolute minimum value is -271 at x = -4.
Therefore, the absolute maximum value of f on the interval [-4, 5] is 98, and the absolute minimum value is -271.