Question

The maximum value of function \(f(x) = 4x^3 - 24x^2 + 36\) in the domain [-1, 5] is ................... (round off to nearest integer).

Hint:

Don't forget to check the endpoints of the interval! A common mistake is to only check the local maxima from the derivative test, but the absolute maximum or minimum on a closed interval can often occur at the boundaries.

Answer 1

Step 1: Understanding the Concept:
To find the absolute maximum value of a continuous function on a closed interval \([a, b]\), we need to evaluate the function at its critical points within the interval and at the endpoints of the interval. The largest of these values will be the absolute maximum.
Step 2: Key Formula or Approach:
1. Find the critical points by taking the first derivative of the function, \(f'(x)\), and setting it to zero.
2. Solve for \(x\) to find the locations of the critical points.
3. Evaluate the function \(f(x)\) at the critical points that lie within the given domain \([-1, 5]\).
4. Evaluate the function \(f(x)\) at the endpoints of the domain, \(x = -1\) and \(x = 5\).
5. Compare all the values obtained in steps 3 and 4 to find the maximum value.
Step 3: Detailed Calculation:
The function is \(f(x) = 4x^3 - 24x^2 + 36\).
The domain is \([-1, 5]\).
1. Find the derivative: \[ f'(x) = \frac{d}{dx}(4x^3 - 24x^2 + 36) = 12x^2 - 48x \] 2. Find the critical points:
Set \(f'(x) = 0\): \[ 12x^2 - 48x = 0 \] \[ 12x(x - 4) = 0 \] The critical points are \(x = 0\) and \(x = 4\).
3. Evaluate \(f(x)\) at critical points within the domain:
Both \(x=0\) and \(x=4\) are within the domain \([-1, 5]\). - At \(x = 0\): \[ f(0) = 4(0)^3 - 24(0)^2 + 36 = 36 \] - At \(x = 4\): \[ f(4) = 4(4)^3 - 24(4)^2 + 36 = 4(64) - 24(16) + 36 = 256 - 384 + 36 = -92 \] 4. Evaluate \(f(x)\) at the endpoints:
- At \(x = -1\): \[ f(-1) = 4(-1)^3 - 24(-1)^2 + 36 = 4(-1) - 24(1) + 36 = -4 - 24 + 36 = 8 \] - At \(x = 5\): \[ f(5) = 4(5)^3 - 24(5)^2 + 36 = 4(125) - 24(25) + 36 = 500 - 600 + 36 = -64 \] 5. Compare the values:
The values we have calculated are:
- \(f(0) = 36\)
- \(f(4) = -92\)
- \(f(-1) = 8\)
- \(f(5) = -64\)
The largest of these values is 36.
Step 4: Final Answer:
The maximum value of the function in the given domain is 36.
Step 5: Why This is Correct:
The solution correctly follows the procedure for finding the absolute maximum of a function on a closed interval. All critical points and endpoints were evaluated, and the maximum value was correctly identified. The answer key range is 36 to 36.