Local min/max of $y =(4+x)/(x^2-5)$

Question

What is the local minimum and maximum of the following function?

$y =(4+x)/(x^2-5)$

From looking at a graph, it seems to me that the local minimum does not exist, and how would I find the maximum?

Answer 1

The derivative will be equal to $$\frac{-x^2 - 8x - 5}{(x^2-5)^2}$$ using Quotient Rule.

Solve the polynomial $-x^2 - 8x - 5$ and then use tests to find min/max.

Answer 2

Be careful when you are looking at the graph. There is both a local minimum and a local maximum.

When you have found the two extrema by using methods described by the others I suggest using the second derivative to check if they are local minima, maxima and/or saddle points. It is safer than just looking at the graph which can sometimes be deceiving. Alternatively you can use a sign table.