find the absolute maximum and minimum values of the following function on the given interval. then graph the…
find the absolute maximum and minimum values of the following function on the given interval. then graph the function.\n\n$f(x)=\\frac{1}{x}+\\ln x, 0.6 \\leq x \\leq 3$\n\nfind the absolute maximum value. select the correct choice below and, if necessary, fill in the answer boxes to complete your choice.\n\na. the absolute maximum value occurs at $x=$\n(type exact answers. use a comma to separate answers as needed.)\n\nb. there is no absolute maximum.
Answer
Explanation:
Step1: Find the derivative of the function
The function is (f(x)=\frac{1}{x}+\ln x), (0.6\leq x\leq3). Using the power rule ((x^n)^\prime = nx^{n - 1}) and ((\ln x)^\prime=\frac{1}{x}), the derivative (f^\prime(x)=-\frac{1}{x^{2}}+\frac{1}{x}=\frac{- 1 + x}{x^{2}}).
Step2: Find the critical points
Set (f^\prime(x) = 0), so (\frac{-1 + x}{x^{2}}=0). Since (x^{2}\gt0) for (x\in[0.6,3]) (except (x = 0) which is not in our domain), then (-1+x=0), and (x = 1).
Step3: Evaluate the function at the critical point and endpoints
- Evaluate (f(x)) at (x=0.6): (f(0.6)=\frac{1}{0.6}+\ln(0.6)=\frac{5}{3}+\ln(0.6)\approx\frac{5}{3}-0.5108\approx1.1559).
- Evaluate (f(x)) at (x = 1): (f(1)=\frac{1}{1}+\ln(1)=1+0 = 1).
- Evaluate (f(x)) at (x=3): (f(3)=\frac{1}{3}+\ln(3)\approx0.3333 + 1.0986=1.4319).
Answer:
A. The absolute maximum value (1.4319) occurs at (x = 3).