Find the indicated absolute extremum as well as all values of x where it occurs on the specified domain
f(x)= x^2e^-0.25x ; [3,10]
Find f'(x) and see if there is a zero (root) on the interval.
If not, the extrema are the two limits of the interval.
Check the sign of f'(x) on the interval. If there is no zero on the interval, the sign should not change (+ OR -).
If f'(x) is + on the interval, it is monotonically increasing, and vice versa.
If it is monotonically increasing or decreasing, then f(c1) and f(c1) are the limits of the range of the function on the given interval [c1,c2]. I.e. the function does not take up values beyond [f(c1),f(c2)].
Post your answer for checking if you wish.
To find the absolute extremum of a function within a specified domain, we can follow these steps:
1. Find the critical points: These are the points where the derivative of the function is either zero or undefined.
2. Evaluate the function at the critical points and the endpoints of the given domain.
3. Identify the maximum and minimum values from the evaluated points.
Let's start by finding the derivative of the function f(x) with respect to x:
f(x) = x^2e^(-0.25x)
To find the derivative, we will use the product rule. Let g(x) = x^2 and h(x) = e^(-0.25x):
f'(x) = g'(x) * h(x) + g(x) * h'(x)
g'(x) = 2x
h'(x) = -0.25e^(-0.25x)
f'(x) = 2x * e^(-0.25x) + x^2 * (-0.25e^(-0.25x))
Now, let's find the critical points by setting f'(x) equal to zero and solving for x:
2x * e^(-0.25x) + x^2 * (-0.25e^(-0.25x)) = 0
Factoring out e^(-0.25x):
e^(-0.25x) * (2x - 0.25x^2) = 0
Setting each factor equal to zero:
e^(-0.25x) = 0 (No solution)
2x - 0.25x^2 = 0
x(2 - 0.25x) = 0
Setting each factor equal to zero:
x = 0 (critical point)
2 - 0.25x = 0
0.25x = 2
x = 8 (critical point)
Now, evaluate the function at the critical points and the endpoints of the domain [3,10]:
f(3) = 3^2 * e^(-0.25*3) ≈ 4.862
f(8) = 8^2 * e^(-0.25*8) ≈ 44.936
f(10) = 10^2 * e^(-0.25*10) ≈ 22.251
By evaluating the function at the critical points and the endpoints, we have the following values:
f(3) ≈ 4.862
f(8) ≈ 44.936
f(10) ≈ 22.251
From the above values, we can see that f(x) has an absolute maximum value of approximately 44.936 at x = 8, and an absolute minimum value of approximately 4.862 at x = 3.