# Solving 3 variable function

I have a function given by

function1[T_,f_,d_] := Module[.......];


I defined the output to be

{f,V(T,f,d)}


A pair of variables which changes everytime I change any of the variables change.

I then proceeded to fix d and T and plot {f,V(f)} on a plot.

The function looks something like this. As you can see the Value of V increases initially before falling off. But this behavior chages. There is a first intervall for T [0,T_1] for which the value increases before falling off just as in the picture. But after a certain treshhold for T the value falls off immediately as seen here.

I want to know for which T the first value for V(T,0,d) is also a maximum such that any later value will be lower

or in other words

I want to find the first T for which the derivative = 0

My reasoning is the following: As long as the derivative is positive the values will always increase a bit. But the moment it becomes 0 I know that this is the point of no return, the threshold I am looking for.

I could do just a analytical derivative. But my professor was talking to my about taylor expanding the funciton around the point 0.

1. Could someone show me how I'd solve a function like this for 0 ?
2. Why would my professor want me to use the taylor expansion when I could just make the derivative myself?
3. Also I seem to have found this treshhold numerically. But there is some strange behavior as the derivative jumps from a positive value to a negative one, no matter how small I make the change in T. I increase T from T= 0.91 to T=0.915 and the derivative jumps from

0.00002 to -0.883796 ´ Could it be that there is no analytical solution where I will be able to find where the derivative is exactly 0?

I spent all day today with this problem and to be honest am quite desperate. Thank you guys in advance