Question

Find the exact absolute maximum and minimum of each function on the stated interval a. h(x)=xe⁻ˣ,[0,3] b. p(t)=sin(t)+cos(t),[−π​/2,π/2​] c. q(x)=x²/x−2​,[3,7] d. f(x)=4−e⁻⁽ˣ⁻²⁾²,(−[infinity],[infinity]) e. h(x)=xe⁻ᵃˣ,[0,2/a​](a>0) f. f(x)=b−e⁻⁽ᵃ⁻ˣ⁾²,(−[infinity],[infinity]),a,b>0

Answer 1

An example solution was worked out for finding the absolute maximum and minimum of the function h(x) = xe^(-x) on the interval [0,3], which are approximately 0.3679 and 0, respectively.

Step-by-step explanation:

The student has asked to find the exact absolute maximum and minimum of several functions over given intervals. Since there are many functions listed, let's focus on solving one of them as an example:
a. h(x)=xe⁻ⁱʸ on the interval [0,3]. To find the absolute maxima and minima, we need to first find the critical points of the function within the interval by taking the derivative and setting it equal to zero. Next, evaluate the function at the critical points and at the endpoints of the interval. The largest value will be the absolute maximum, and the smallest will be the absolute minimum.

Let's differentiate the function h(x):

h'(x) = e⁻ⁱʸ - xe⁻ⁱʸ
Setting h'(x) to zero gives: e⁻ⁱʸ(1 - x) = 0.
From this, we find that x = 1 is a critical point. Evaluating h(x) at x = 0, x = 1, and x = 3 gives:

• h(0) = 0 ⋅ e⁻⁰ = 0
• h(1) = 1 ⋅ e⁻¹ ≈ 0.3679
• h(3) = 3 ⋅ e⁻³ ≈ 0.1494

Thus, the absolute maximum is approximately 0.3679 at x = 1, and the absolute minimum is 0 at x = 0 on the interval [0,3].