# The graph doesn't generate

I try graph the partial derivative of this function but it keeps giving me the "invalid variable". What I am trying to follow is: 1. Take the derivative of the function respect to T only (To and Eo is constant), then I solve for T by letting the df/dT = 0 and graph the final result. Can someone point out what did I miss or make error on? ( I'm a beginning Mathematica selflearner). I really appreciate your help. Thank you.

• hello, could you supply the values for the constants T0 and Eo? Commented Dec 11, 2019 at 21:56
• The question doesn't give me the exact number, it's said "Eo and To are constants" Commented Dec 11, 2019 at 22:09
• I solved to get the Cv maximum but then I'm stuck with plot the graph in the end: I get {{T -> -3.23996 To}, {T -> 0.411528 To}} as the answer. Commented Dec 11, 2019 at 22:50
• This not a well posted question. What is Cv? How is it related to f?. E0 is just a scale factor; sitting it to 1 would simplify your question without distorting it. Commented Dec 11, 2019 at 23:53

## 1 Answer

This is not an answer. As the question is currently posed it can not be answered even if one corrects the syntax errors in post. This post is to show why that is so.

Since E0 is just a scale factor, I will assume E0 = 1. This will not invalidate my argument.

So consider

f[t0_][t_] = With[{u = t/t0 + Sqrt[2]}, u^-4 - u^-2]

(4/((Sqrt[2] + t/t0)^5 t0)) + 2/((Sqrt[2] + t/t0)^3 t0)


Then the derivative is

df[t0_][t_] = D[f[t0][t], t]

-(4/((Sqrt[2] + t/t0)^5 t0)) + 2/((Sqrt[2] + t/t0)^3 t0)


What are some of salient characteristics of df?

df[t0][0]


0

So, when t0 != 0, there is a zero at t = 0.

Now consider

Reduce[df[t0][t] > 0 && t > 0 && t0 > 0, {t, t0}]


t > 0 && t0 > 0

This means, for any t0 > 0, df is always positive and has no zeros in for t > 0. Therefore, solving for zeros of df in the domain (0, 50] is doomed to failure. It also turns out that f is a function the grows quickly near zero and becomes asymptotic to the x-axis as t approaches infinity.

I offer this Manipulate to anyone who wants to visualize what has found analytically above.

Manipulate[
Column[{
Style["Function", "SR"],
Plot[f[t0][t], {t, 1, tmax}, PlotRange -> All, ImageSize -> Medium],
Style["Derivative", "SR"],
Plot[df[t0][t], {t, 1, tmax}, PlotRange -> All, ImageSize -> Medium]
}],
{tmax, 50, 1000, 50, Appearance -> "Labeled"},
{t0, 5, 100, 5, Appearance -> "Labeled"}]


demo