Consider the function f(x)=(lnx)/(x) shown here: (a) Find the average value of f(x) on [(1)/(2),

step 1 Okay, so for the problem, we're going to use this a formula for the average value, and we're jus

Consider the function f(x)=(lnx)/(x) shown here: (a) Find...

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

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Key Concepts

Average Value of a Function

The average value of a function over a closed interval is defined as the integral of the function over that interval divided by the measure (length) of the interval. This concept is fundamental in understanding how a function behaves on average across a specified range and is widely used in various applications of calculus, including physics and statistics.

Antiderivatives and Definite Integrals

Antiderivatives are used to evaluate definite integrals, which represent the accumulation of quantities. In the context of finding an average value, calculating the definite integral of the function over the given interval is essential because it captures the total effect of the function across that interval. Proper integration techniques help in simplifying these calculations.

Mean Value Theorem for Integration

The Mean Value Theorem for Integration guarantees that, for a continuous function on a closed interval, there exists at least one point in the interval where the function's value equals its average value over that interval. This theorem is key to establishing the existence of a specific point (often denoted as 'c') where the function exactly represents the average behavior observed over the entire interval.

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