Determine the absolute extrema of each function on the given interval. Illustrate your results by
sketching the graph of each function.
a) ���� = �� − 4� + 3 , 0 ≤ � ≤ 3
b) ���� = �� − 1�� , 0 ≤ � ≤ 2
To determine the absolute extrema of a function on a given interval, you need to find the maximum and minimum values of the function within that interval. Here's how you can do it for each function:
a) To find the absolute extrema of the function f(x) = x^3 - 4x + 3 on the interval 0 ≤ x ≤ 3, follow these steps:
1. Find the critical points of the function by taking its derivative and solving for f'(x) = 0.
f'(x) = 3x^2 - 4 = 0
Solving this equation, we get x = ± √(4/3).
However, we only need to consider the critical points that fall within the given interval.
2. Evaluate the function at the critical points and the endpoints of the interval.
f(0) = 3
f(√(4/3)) ≈ 0.33
f(3) = 18
3. Compare the function values and determine the maximum and minimum values.
The maximum value is the highest function value obtained, which is 18 at x = 3.
The minimum value is the lowest function value obtained, which is 0.33 at x = √(4/3).
Now, you can sketch the graph of the function f(x) = x^3 - 4x + 3 using these results.
b) Similarly, to find the absolute extrema of the function g(x) = x^2 - x^3 on the interval 0 ≤ x ≤ 2, follow these steps:
1. Find the critical points of the function by taking its derivative and solving for g'(x) = 0.
g'(x) = 2x - 3x^2 = 0
Solving this equation, we get x = 0 and x ≈ 0.67.
Again, consider only the critical points within the given interval.
2. Evaluate the function at the critical points and the endpoints of the interval.
g(0) = 0
g(0.67) ≈ 0.07
g(2) = -4
3. Compare the function values and determine the maximum and minimum values.
The maximum value is the highest function value obtained, which is 0 at x = 0.
The minimum value is the lowest function value obtained, which is -4 at x = 2.
Now, you can sketch the graph of the function g(x) = x^2 - x^3 using these results.