NIntegrate allegedly evaluating to non-numerical values

Question

I'm getting non-numerical-evaluation error messages despite not having any symbolic variables in my code.

Here's a MWE representative of what I'm doing:

f[x_,y_]:=NIntegrate[(2y-(x-a/2)^2),{a,0,1}];
FindMaximum[{f[x,0.5],0<x<1},{x,0}]

This results in the following error:

NIntegrate: The integrand 1.-((-(a/2)+x)^2) has evaluated to non-numerical values for all sampling points in the region with boundaries {{0,1}}.

I don't see what the problem is. $f$ is clearly a function of $x$ and $y$ and has no non-numerical parameters. I fact, it seems to behave properly other than when I ask it to find the maximum:

f[0.2,0.5]

0.9767

Plot[f[x,0.5],{x,0,1}]

I've tried adding ?NumericQ in the function definition as follows, but I get the same result:

f[x_,y_?NumericQ]:=NIntegrate[(2y-(x-a/2)^2),{a,0,1}];

I've also tried adding ?NumericQ to to function arguments when I call the function in the FindMaximum command, again to no avail:

FindMaximum[{f[x,0.5?NumericQ],0<x<1},{x,0}]

I had a look at this Q&A, but my situation doesn't fall into any of the cases described there.

What's going on?

Answer 1

The problem is that Mathematica tries to evaluate f[x,0.5] before x is defined. This issue is explained clearly in this Wolfram support page.

To force the function to only evaluate if it receives numerical values for both x and y:

Clear[f];
f[x_?NumericQ, y_?NumericQ] := NIntegrate[(2 y - (x - a/2)^2), {a, 0, 1}]
FindMaximum[{f[x, 0.5], 0 < x < 1}, {x, 0}]
 (* {0.979167, {x -> 0.25}} *)

In this particular case, you really only need to restrict x:

Clear[f];
f[x_?NumericQ, y_] := NIntegrate[(2 y - (x - a/2)^2), {a, 0, 1}]