Problem 44 - Essential Calculus

Question

Find the average value of the function over the given interval and all values of \(x\) in the interval for which the function equals its average value. $$ f(x)=\frac{4\left(x^{2}+1\right)}{x^{2}}, \quad[1,3] $$

Step-by-Step Solution

Answer

The average value of the function over the given interval is \(2 - 2 \log 3\). Now, by equating this value to the function and simplifying, we can find the specific values of \(x\) in the interval [1, 3] for which the function equals its average value. This part will need some algebraic manipulation and possibly the use of a numerical method.

Step 1: Compute the average value of function f(x)

The average value of \( f(x) \) over the interval \([a, b]\) is given by the formula: \[ \textup{Average value} = \frac{1}{b - a} \int_{a}^{b} f(x) dx \] Therefore, the average value of the given function \(f(x) = \frac{4\left(x^{2}+1\right)}{x^{2}} \) over the interval \([1,3]\) is computed as follows: \[ \textup{Average value} = \frac{1}{3 - 1} \int_{1}^{3} \frac{4\left(x^{2}+1\right)}{x^{2}} dx = \frac{1}{2} \left[\frac{4x}{x^{2}} + 4\log |x|\right]_1^3 = \frac{1}{2}[(8 - 4\log |1|) - (4 - 4\log |3|)] = 2 - 2 \log 3 \]

Step 2: Find values of x for which f(x) equals its average value

To find values of \(x\) for which \(f(x)\) equals its average value, we should equate \(f(x)\) to the computed average value and solve for \(x\): \[ \frac{4\left(x^{2}+1\right)}{x^{2}} = 2 - 2 \log 3 \] Simplify and solve the above equation for \(x\) and only consider values in the interval [1, 3]. This step involves some algebraic manipulations and possibly the use of a numerical method to find the roots.