Solution Found!
x A family of super-exponential functions Let f(x) = (a
Chapter 4, Problem 76RE(choose chapter or problem)
A family of super-exponential functions Let \(f(x)=(a+x)^x\), where a > 0.
a. What is the domain of f (in terms of a)?
b. Describe the end behavior of f (near the boundary of its domain or as \(|x| \rightarrow \infty\)).
c. compute f’. Then graph f and f’ for a = 0.5, 1, 2, 3.
d. Show that f has a single local minimum at the point z that satisfies (z+a) ln (z+a) + z = 0.
e. Describe how z [found in part (d)] varies as a increases. Describe how f(z) varies as a increases.
Questions & Answers
QUESTION:
A family of super-exponential functions Let \(f(x)=(a+x)^x\), where a > 0.
a. What is the domain of f (in terms of a)?
b. Describe the end behavior of f (near the boundary of its domain or as \(|x| \rightarrow \infty\)).
c. compute f’. Then graph f and f’ for a = 0.5, 1, 2, 3.
d. Show that f has a single local minimum at the point z that satisfies (z+a) ln (z+a) + z = 0.
e. Describe how z [found in part (d)] varies as a increases. Describe how f(z) varies as a increases.
ANSWER:Solution 76RE Step 1 In this problem we are given that f(x) = (a+x) where a > 0. We have to find the domain of f in terms of a and we have to explain the end behavior of f near the boundary. Then we need to find f and we need to draw the graph of f and f . Then we have to show that f has only one local minimum at the point z that satisfies the given condition.