Find the absolute maximum and minimum values of the function f(x)=3(x-2)^(2/3)+5 on the interval [1,10]?
I'm not sure how to do this. Please help.
to find the absolute max/min, you need to check for local extrema in the interval. Then check f(x) at the ends of the interval to see whether they qualify as being more max/min.
f'(x) = 2/∛(x-2)
clearly that is never zero, so there are no internal local extrema.
So, now just check f(1) and f(10). They will be the absolute min and max.
To find the absolute maximum and minimum values of a function on a closed interval, you can follow these steps:
1. Determine the critical points: Find the values of x where the derivative of the function is either zero or undefined. In this case, you have a function f(x) = 3(x-2)^(2/3) + 5.
First, calculate the derivative of f(x) using the Chain Rule:
f'(x) = 2(x-2)^(-1/3) * 3/3 = 2(x-2)^(-1/3).
Next, set f'(x) equal to zero and solve for x:
2(x-2)^(-1/3) = 0.
Since the derivative is never undefined for this function, the only possible critical point occurs when (x-2)^(-1/3) = 0. This happens when x = 2.
2. Evaluate the function at the endpoints: Plug in the values of the endpoints (1 and 10) into the function f(x) = 3(x-2)^(2/3) + 5.
f(1) = 3(1-2)^(2/3) + 5 = 3(-1)^(2/3) + 5 = 3(-1) + 5 = 2.
f(10) = 3(10-2)^(2/3) + 5 = 3(8)^(2/3) + 5.
3. Evaluate the function at the critical point(s): Plug in the value(s) of x where the derivative is zero (in this case, just x = 2) into the function f(x).
f(2) = 3(2-2)^(2/3) + 5 = 3(0)^(2/3) + 5 = 3(0) + 5 = 5.
4. Compare the values obtained in steps 2 and 3: Determine which value is the absolute maximum and which is the absolute minimum on the interval [1,10].
In this case, f(1) = 2, f(2) = 5, and the value of f(10) needs to be calculated. Calculate f(10) using the formula mentioned earlier.
f(10) = 3(8)^(2/3) + 5.
Now, compare the three values: f(1), f(2), and f(10).
The absolute maximum value is the largest value among these three, and the absolute minimum value is the smallest.
Therefore, the absolute maximum value is f(10) (which you need to calculate) or 5 if f(10) is smaller than 5.
The absolute minimum value is f(1), which is 2.