# Thread: Minimum and Maximum points

1. ## Minimum and Maximum points

Okay, I'm solving to find the min and max of a function, f(x)= (a*x)/(x^2 + a^2). I found the derivative, set it to zero and concluded that the critical points are +/-a. So now I have to take a value a bit less than a and a value a bit greater than a to find out if the point is a min or max point. How do I do this if I have no numerical values for x or a? Can someone put some perspective on this? Thanks so much!

2. Originally Posted by MrJoe2000
Okay, I'm solving to find the min and max of a function, f(x)= (a*x)/(x^2 + a^2). I found the derivative, set it to zero and concluded that the critical points are +/-a. So now I have to take a value a bit less than a and a value a bit greater than a to find out if the point is a min or max point. How do I do this if I have no numerical values for x or a? Can someone put some perspective on this? Thanks so much!
You can use the 2nd derivative test to see if a point is a max or a min.

If a is a critical point of f then if

$\displaystyle f''(a)< 0 \implies \text{Max} \quad f''(a)> 0 \implies \text{Max} \quad f''(a)= 0 \implies \text{saddle point}$