Let f(x)=x^3-3x^2 find the absolute maximum value of f(x) on the interval [1,4]?
22 hours ago - 3 days left to answer
f' = 3 x^2 - 6 x = 0 at max or min
x(3x-6) = 0
x = 0 or 2 but 0 is outside interval
f" = 6 x - 6
at x = 2, f" = +6
so that is a minimum, not a max
therefore look at the end points
at x = 1, f(1) = 1-3 = -2
at x = 4, f(4) = 16
so
minimum at x = 1
To find the absolute maximum value of f(x) on the interval [1,4], we can follow these steps:
1. Calculate the critical points of f(x) by finding where the derivative equals zero or is undefined.
- Find the derivative of f(x): f'(x) = 3x^2 - 6x.
- Set f'(x) equal to zero and solve for x: 3x^2 - 6x = 0.
- Factor out x: x(3x - 6) = 0.
- Set each factor equal to zero: x = 0 or 3x - 6 = 0.
- Solve for x: x = 0 or x = 2.
2. Evaluate the function f(x) at the critical points and the endpoints of the interval [1,4] to determine the maximum value.
- Calculate f(0): f(0) = (0)^3 - 3(0)^2 = 0.
- Calculate f(1): f(1) = (1)^3 - 3(1)^2 = -2.
- Calculate f(2): f(2) = (2)^3 - 3(2)^2 = -4.
- Calculate f(4): f(4) = (4)^3 - 3(4)^2 = -32.
3. Compare the values obtained in step 2 to find the absolute maximum value.
- The maximum value of f(x) on the interval [1,4] is the largest value obtained from step 2.
- In this case, the largest value is f(1) = -2.
Therefore, the absolute maximum value of f(x) on the interval [1,4] is -2.
To find the absolute maximum value of a function on a given interval, you need to evaluate the function at its critical points and endpoints within that interval. Here's how you can do it for the function f(x) = x^3 - 3x^2 on the interval [1,4]:
1. Find the critical points of the function within the given interval. Critical points occur where the derivative of the function is either zero or undefined. To find the derivative of f(x), differentiate it with respect to x:
f'(x) = 3x^2 - 6x
Next, set the derivative equal to zero and solve for x:
3x^2 - 6x = 0
3x(x - 2) = 0
This equation has two solutions: x = 0 and x = 2. However, we need to confirm if these critical points lie within the interval [1,4].
2. Evaluate the function f(x) at the critical points and endpoints. The function values at the critical points and endpoints will help identify the absolute maximum. So, calculate the following values:
f(1) = (1)^3 - 3(1)^2 = 1 - 3 = -2
f(4) = (4)^3 - 3(4)^2 = 64 - 48 = 16
f(0) = (0)^3 - 3(0)^2 = 0
We do not need to calculate f(2), as it is not within the given interval.
3. Compare the function values calculated in step 2 to determine the absolute maximum.
The function values within the interval [1,4] are f(1) = -2 and f(4) = 16. Since 16 is greater than -2, the absolute maximum value of f(x) on the interval [1,4] is 16.
Hence, the absolute maximum value of f(x) on the interval [1,4] is 16.