Question

question 20 of 20
identify the relative maximum value of g(x) for the function shown below.
( g(x) = \frac{2}{x^2 + 3} )

Explanation:

Step1: Analyze the denominator's range

The denominator of the function \(g(x)=\frac{2}{x^{2}+3}\) is \(x^{2}+3\). Since \(x^{2}\geq0\) for all real \(x\), then \(x^{2}+3\geq3\) (by adding 3 to both sides of the inequality \(x^{2}\geq0\)).

Step2: Analyze the function's range based on the denominator

We know that for a fraction \(\frac{a}{b}\) (where \(a = 2>0\) here), as \(b\) increases, the value of the fraction decreases, and as \(b\) decreases, the value of the fraction increases. Since \(x^{2}+3\geq3\), the reciprocal of \(x^{2}+3\), which is \(\frac{1}{x^{2}+3}\), satisfies \(0<\frac{1}{x^{2}+3}\leq\frac{1}{3}\) (because when the denominator is larger, the fraction is smaller, and the maximum value of \(\frac{1}{x^{2}+3}\) occurs when the denominator is minimized, i.e., when \(x = 0\) and the denominator is 3).
Then, multiplying each part of the inequality \(0<\frac{1}{x^{2}+3}\leq\frac{1}{3}\) by 2 (since 2 is a positive constant, the inequality direction remains the same), we get \(0<\frac{2}{x^{2}+3}\leq\frac{2}{3}\).

Step3: Determine the relative maximum

The function \(g(x)=\frac{2}{x^{2}+3}\) reaches its maximum value when the denominator \(x^{2}+3\) is minimized. The minimum value of \(x^{2}+3\) occurs at \(x = 0\) (because \(x^{2}\) is minimized at \(x = 0\) with \(x^{2}=0\)), and substituting \(x = 0\) into \(g(x)\), we get \(g(0)=\frac{2}{0^{2}+3}=\frac{2}{3}\). Since this is the maximum value the function can take (from the range analysis above), this is the relative maximum (and also the absolute maximum in this case).

Answer:

\(\frac{2}{3}\)