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 $\\square$ occurs at $x=\\square$\n(type exact answers. use a comma to separate answers as needed)\n\nb. there is no absolute maximum.

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 $\\square$ occurs at $x=\\square$\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), its 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), then (\frac{-1 + x}{x^{2}}=0). Since (x^{2}>0) for (x\in[0.6,3]) (except (x = 0) which is not in our domain), we solve (-1+x=0), getting (x = 1)

Step3: Evaluate the function at critical points and endpoints

  • Evaluate (f(x)) at (x=0.6): (f(0.6)=\frac{1}{0.6}+\ln(0.6)=\frac{5}{3}+\ln\frac{3}{5}\approx\frac{5}{3}- 0.5108\approx1.6667-0.5108 = 1.1559)
  • 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:

The absolute maximum value (1.4319) occurs at (x = 3)