 Hello friends and how are you all today the question says find the intervals on which the function fx equal to x upon 1 plus x square is Increasing and then decreasing that is rewrite the given function once again We have fx equal to x upon 1 plus x square now. Let us find out the derivative of this function So the derivative of this function on applying quotient rule will be 1 plus x square into derivative of numerator minus x into derivative of the denominator Upon denominators whole Square that further implies that we have f dash x equal to 1 plus x square minus 2 x square On 1 plus x square the whole square That gives us f dash x equal to 1 minus x square on 1 plus x square the whole Square now here the nature of this derivative depends on the numerator that is 1 minus x square as 1 plus x 1 plus x square the whole square that is the denominator is always positive as a square of some number is always Positive now. Let us find out the value of x by Putting f dash x equal to 0 this gives us 1 minus x square is equal to 0 So 1 plus x into 1 minus x is equal to 0 that implies x is equal to minus 1 or x is equal to 1 so we have the intervals as minus infinity to minus 1 then minus 1 To 1 1 to infinity Now in these three intervals we will be finding out whether f dash x is less than 0 greater than 0 So you found out that in the interval minus infinity to minus 1 the value of f dash x is less than zero in the next Interval it's greater than the third interval again. It's less than zero. So this implies that the function which is given to us is increasing in the interval minus 1 to 1 and is decreasing in the interval first of all minus infinity to minus 1 and then 1 to Infinity right so this is the answer to the given question So hope you understood the concept well and enjoyed it to have a nice day