# Why is Mathematica returning the original function from FindMaximum? [closed]

I am trying to find the maximum of the following expression:

I can not figure out why Mathematica returns me the original expression.

I am new to Mathematica and tired to search around but could not find an answer to address this question.

Code corresponding to the image posted above:

FindMaximum[
{(p (4 t^2 V (-1 + α) α - 4 p^3 β2 +
p^2 (β1^2 + β2 + 4 V β2 - 4 t (α - β2 + α β2)) -
p t (4 t α^2 + β1^2 + 4 V β2 - α (1 + 4 t + β1^2 + 4 V (1 + β2))))) /
(4 t (p + t (-1 + α)) (t α + p β2)),
0 <= α <= 1},
{α, 0}]


## closed as off-topic by b3m2a1, Roman, MarcoB, user42582, Alex TrounevJun 3 at 16:17

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• "This question arises due to a simple mistake such as a trivial syntax error, incorrect capitalization, spelling mistake, or other typographical error and is unlikely to help any future visitors, or else it is easily found in the documentation." – b3m2a1, Roman, MarcoB, user42582, Alex Trounev
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• Please always post copyable code, not just a screenshot. We can't run your code. – Szabolcs May 29 at 20:43
• Note that FindMaximum uses numerical methods, but you gave it an expression with symbolic parameters. It can't possibly work. I don't immediately see why it didn't show an error though. – Szabolcs May 29 at 20:44
• added the code. If I wish to contain the symbolic parameters and find a maximum of this function, but based on the first order and second order derivative, it is not immediately obvious whether by setting FOC = 0 will return a maximum value. In this case, what should I do then? – lll May 29 at 20:52

there are two candidates for the maximum, just check those:

f[α_] = (p (4 t^2 V (-1 + α) α - 4 p^3 β2 + p^2 (β1^2 + β2 + 4 V β2
- 4 t (α - β2 + α β2)) - p t (4 t α^2 + β1^2 + 4 V β2
- α (1 + 4 t + β1^2 + 4 V (1 + β2)))))/(4 t (p + t (-1 + α))(t α
+ p β2));

f'[α] // FullSimplify


1/4 p^2 (-(1/(p + t (-1 + α))^2) - β1^2/(t α + p β2)^2)

sol = Solve[% == 0, α] // FullSimplify


$$\left\{\left\{\alpha \to -\frac{\sqrt{\beta_1^2 \left(-t^2\right) ((\beta_2-1) p+t)^2}+\beta_1^2 t (p-t)+\beta_2 p t}{\left(\beta_1^2+1\right) t^2}\right\},\left\{\alpha \to \frac{\sqrt{\beta_1^2 \left(-t^2\right) ((\beta_2-1) p+t)^2}-p t \left(\beta_1^2+\beta_2\right)+\beta_1^2 t^2}{\left(\beta_1^2+1\right) t^2}\right\}\right\}$$

There are four candidates for the maximum: $$\alpha=0$$, $$\alpha=1$$, and the two solutions above:

A = {0, 1, α /. sol[[1]], α /. sol[[2]]};


For given parameters $$\beta_1$$, $$\beta_2$$, $$p$$, $$t$$ you need to first check if the last two of these lie in the range $$[0,1]$$, and then compare the values of the function at these points to see which one is the largest:

FullSimplify[f /@ A]
(* lengthy output of four values of the function *)


If you give example values of the parameters $$\beta_1$$, $$\beta_2$$, $$p$$, $$t$$, I can show you an example.

• thanks, but I am a bit confused here. Is that for solving for maximum within a specified range of a variable (like the function I have above), setting FOC == 0, and check for all the values with FOC == 0, and corner points? Thanks for the clarification! – lll May 29 at 22:23
• What does "FOC" stand for? – Roman May 30 at 8:47
• first order condition – lll May 30 at 13:27
• Yes that sounds like it then. Just give it a try with concrete parameters. – Roman May 30 at 13:39