Chapter 5: Q. 81 (page 418)

Consider the function $f (x) = \frac{\ln x}{x}$ shown here:
(a) Find the average value of $f (x)$ on role="math" localid="1649276701979" $[\frac{1}{2}, 2]$ .
(b) Find a value $c \in [\frac{1}{2}, 2]$ at which $f (x)$ achieves its average value.

Short Answer

Expert verified

Part (a) The value is $0 .$

Part (b) The value is $c = 1 .$

Step by step solution

Part (a) Step 1. Given information.

The given function is $f (x) = \frac{\ln x}{x}$

Part (a) Step 2. Explanation.

The average value of the function will be,

$Avg. = \frac{1}{b - a} \int_{a}^{b} f (x) d x = \frac{1}{2 - \frac{1}{2}} \int_{\frac{1}{2}}^{2} \frac{\ln x}{x} d x = \frac{2}{3} {[\frac{1}{2} (\ln (x))^{2}]}_{\frac{1}{2}}^{2} + c = \frac{2}{3} [\frac{1}{2} [(\ln (2))^{2}] - {(\ln (\frac{1}{2}))}^{2}] + c = 0$

Part(b) Step 1. Explanation.

Now,

c = 1, f (x) = \frac{\ln 1}{1} = 0

Therefore,

The value of $c = 1 .$

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Consider the function $f (x) = \frac{\ln x}{x}$ shown here:
(a) Find the average value of $f (x)$ on role="math" localid="1649276701979" $[\frac{1}{2}, 2]$ .
(b) Find a value $c \in [\frac{1}{2}, 2]$ at which $f (x)$ achieves its average value.

Short Answer

Step by step solution

Part (a) Step 1. Given information.

Part (a) Step 2. Explanation.

Part(b) Step 1. Explanation.

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Consider the function f(x)=lnxxshown here: (a) Find the average value off(x) on role="math" localid="1649276701979" 12,2. (b) Find a value c∈12,2 at which f(x) achieves its average value.

Short Answer

Step by step solution

Part (a) Step 1. Given information.

Part (a) Step 2. Explanation.

Part(b) Step 1. Explanation.

One App. One Place for Learning.

Most popular questions from this chapter

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Consider the function $f (x) = \frac{\ln x}{x}$ shown here:
(a) Find the average value of $f (x)$ on role="math" localid="1649276701979" $[\frac{1}{2}, 2]$ .
(b) Find a value $c \in [\frac{1}{2}, 2]$ at which $f (x)$ achieves its average value.