Differentiation; Maxima and Minima

Question

Suppose $y=\frac{1}{x(1+n^{2}x^{2})}$ . Then when will y be maximum and what its maximum value. I don't know how to differentiate it so that i can find the maximum value. Please help me.

Answer 1

I assume the variable is $x$, and $n$ is any real constant (an integer will do, but the following is valid for any real $n$).

If $x$ is free to take any value in $]-\infty,0[\,\cup\,]0,+\infty[$ (as $y$ is not defined for $x=0$), then there is no minimum value. Just pick an arbitrary small positive number $\epsilon$, and let $x=-\epsilon$, then $y$ takes on arbitrarily large negative values. There is no infimum in the reals, nor a minimum, since the set of values of $y$ has no lower bound.

If $x$ is restricted to positive values, that is $x>0$, then there is no minimum either, but there is an infimum: let $x$ be an arbitrarily large positive number, then $y$ takes on arbitrarily small positive values, but never reaches $0$, so the inf is zero, but it is not reached so the minimum does not exist.

Last, there is no maximum. Let $x=\epsilon$, then $y$ takes on arbirarily large positive values. There is no supremum in the reals, nor a maximum, since the set of values of $y$ has no upper bound.

Worth reading: https://en.wikipedia.org/wiki/Infimum_and_supremum

Answer 2

HINT

If u,v are functions of x and when v/v is a maximum or constant, applying quotient rule

$$ \frac{u}{v}= \frac{v du - u dv} {v^2}=0 $$

gives

$$ \frac{u}{v}= \frac{du/dx} {dv/dx} $$

or

$$ \frac{1}{x(1+ n^2 x^2)}= \frac{0} {1+ 3 n^2 x^2} $$

on simplification

$$ 1+ 3 n^2 x^2 =0 $$

which can have no real roots.No max/min for a monotonic function.