Question

Find the absolute max/min values of f(x) = x/x²+1 on the interval [-2,2].

Answer 1

We have the function:

Rewrite it as:

Differentiate above equation with respect to x:

Use product rule:

Now differentiate using the following formulae:

1

2. Chain rule

We get:

Now equate it to 0 to find critical point:

Now We can perform the second derivative test to see maxima or minima lies at which critical point. But in this question we can just put the values of these critical points and the end points in the function to check the maxima/ minima.

Firstly put critical points in the function f(x):

Put

Put

Firstly put end points in the function f(x):

Put

Put

So now we can check from the above values that absolute maxima is 0.5 and absolute minima is -0.5.

Result can be checked in plot below: