If f(x) = 1/x and g(x) = x/(x^2 + 1), for all x > 0, what is the minim

Nested Functions

\(f(x)=\frac{1}{x}\) and \(g(x) = \frac{x}{(x^2+1)}\) and we need to find the minimum value of \(f(g(x))\)

To find \(f(g(x))\) we need to substitute value of g(x) first. So lets do that

\(f(g(x))\) = \(f(\frac{x}{(x^2+1)})\)

To find value of \(f(\frac{x}{(x^2+1)})\) we need to replace x with \(\frac{x}{(x^2+1)}\) in f(x)

We get, \(f(\frac{x}{(x^2+1)})\) = 1 / \(\frac{x}{(x^2+1)}\) = \(\frac{(x^2+1)}{x}\) = \(\frac{x^2}{x}\) + \(\frac{1}{x}\) = x + \(\frac{1}{x}\)

Now we can differentiate and get the answer but let's go with substitution.
x + \(\frac{1}{x}\) = 0 is not possible as x > 0

Now for any value of x > 1, like x = 1.1
x + \(\frac{1}{x}\) will be more than 2

Any value of x between 0 and 0.5 again, like x = 0.1
x + \(\frac{1}{x}\) will be more than 2

For values of x between 0.5 and 1, like x=0.6
x + \(\frac{1}{x}\) will be more than 2

So, Minimum value of x + \(\frac{1}{x}\) will be 2 when x = 1

So, Answer will be E
Hope it helps!

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