To find the absolute maximum and minimum values of a function on a given interval, we need to evaluate the function at its critical points and endpoints, and compare the function values.

Step 1: Find the derivative of the function f(x) = 3 + 81x - 3x^3.

f'(x) = 81 - 9x^2

Step 2: Set the derivative equal to zero and solve for x to find the critical points.

81 - 9x^2 = 0
9x^2 = 81
x^2 = 9
x = -3, 3

However, only x = 3 is in the interval [0, 4].

Step 3: Evaluate the function at the critical points and endpoints.

f(0) = 3 + 81(0) - 3(0)^3 = 3
f(3) = 3 + 81(3) - 3(3)^3 = 3 + 243 - 81 = 165
f(4) = 3 + 81(4) - 3(4)^3 = 3 + 324 - 192 = 135

Step 4: Compare the function values.

The absolute maximum value is 165 at x = 3, and the absolute minimum value is 3 at x = 0.

Question

To find the absolute maximum and minimum values of a function on a given interval, we need to evaluate the function at its critical points and endpoints, and compare the function values.

Step 1: Find the derivative of the function f(x) = 3 + 81x - 3x^3.

f'(x) = 81 - 9x^2

Step 2: Set the derivative equal to zero and solve for x to find the critical points.

81 - 9x^2 = 0
9x^2 = 81
x^2 = 9
x = -3, 3

However, only x = 3 is in the interval [0, 4].

Step 3: Evaluate the function at the critical points and endpoints.

f(0) = 3 + 81(0) - 3(0)^3 = 3
f(3) = 3 + 81(3) - 3(3)^3 = 3 + 243 - 81 = 165
f(4) = 3 + 81(4) - 3(4)^3 = 3 + 324 - 192 = 135

Step 4: Compare the function values.

The absolute maximum value is 165 at x = 3, and the absolute minimum value is 3 at x = 0.

Knowee AI · Accepted Answer

To find the absolute maximum and minimum values of a function on a given interval, we need to evaluate the function at its critical points and endpoints, and compare the function values.

Step 1: Find the derivative of the function f(x) = 3 + 81x - 3x^3.

f'(x) = 81 - 9x^2

Step 2: Set the derivative equal to zero and solve for x to find the critical points.

81 - 9x^2 = 0
9x^2 = 81
x^2 = 9
x = -3, 3

However, only x = 3 is in the interval [0, 4].

Step 3: Evaluate the function at the critical points and endpoints.

f(0) = 3 + 81(0) - 3(0)^3 = 3
f(3) = 3 + 81(3) - 3(3)^3 = 3 + 243 - 81 = 165
f(4) = 3 + 81(4) - 3(4)^3 = 3 + 324 - 192 = 135

Step 4: Compare the function values.

The absolute maximum value is 165 at x = 3, and the absolute minimum value is 3 at x = 0.

Find the absolute maximum and absolute minimum values of f on the given interval.f(x) = 3 + 81x − 3x3, [0, 4]absolute minimum value absolute maximum value

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