Error with fixes

Let's go in parts:

int arr[MAX];
int mayor=arr[0], menor=arr[0];

At this time, the content of arr[0] is unknown. It can be anything, because you have not initialized the array.

Since we do not know that value, it turns out that mayor is already greater than any of the elements you put (and would not change in the iterations). O menor is less than any of your elements (and would not change in iterations).

For example, if menor starts with 0, none of the elements in your example would replace it. menor has to be so high that you are sure that you will find a lower value, mayor so low that you are sure that you will find a higher value.

int diferencia=0;
int main() {
  for(int i=1; i<=10; i++)
  {
    cin>>arr[i];

If you do not need to save the data, you really do not need an array. If you make int numero; cin >> numero; it works exactly the same.

Naturally, in the following lines you would have to change arr[i] by numero .

    if(arr[i]>mayor) {
        mayor=arr[i];
    }
    if(arr[i]<menor);

You have a ; , so the if ends on the top line and the block below always runs.

    {
        menor=arr[i];
    }
}
cout<<menor<<" "<<mayor;

The impression is complicated because it is confused with the data of the last line. I think that's part of the problem; it's much better: cout << endl << "Menor :" << menor << endl << "Mayor: " << mayor << endl;

}

When you start to program it is very normal that some details escape but once you have time you notice certain details, your mistake is that you put a ; where you should not go.

  #include<iostream>
  #define MAX 10002
  using namespace std;

  int arr[MAX];
  int mayor=arr[0], menor=arr[0];
  int diferencia=0;

int main()
{
    for(int i=1; i<=10; i++)
    {
        cin>>arr[i];

        if(arr[i]>mayor)
        {
            mayor=arr[i];
        }

        if(arr[i]<menor); <---- precisamente aquí
        {
           menor=arr[i];
        }
    }
    cout<<menor<<" "<<mayor;
}

Basically the program has two errors. One has already been indicated and it is that you have cast a ; after a if :

if(arr[i]<menor);
//              ^

That semicolon causes the if to do absolutely nothing ... silly language details. To see it more clearly we could replace that semicolon with two keys:

if(arr[i]<menor){ }
{
  menor = arr[i];
}

Now it looks better than that semicolon overrides the conditional.

On the other hand the initialization of menor is not correct.

Global arrays are initialized automatically, not those that belong to a function. That is:

int arr[MAX]; // Todos sus elementos valen 0
              // Pero únicamente porque es variable global!!!

int main()
{
  int arr2[MAX]; // Array no inicializado!!!
}

Well, the next thing you do after declaring the fix is to declare and initialize mayor and menor :

int mayor=arr[0], menor=arr[0];

And here the problems begin ... this initialization makes mayor=menor=0 . If we subsequently give values to arr[0] , the value of mayor and menor will not change automatically. Your starting value will be always 0.

This initialization poses a problem and is that for menor find the smallest number, it must be equal to or less than 0. Otherwise the program will not work.

To avoid this mess of minimums and maximums I would use pointers:

int* mayor = arr;
int* menor = arr;

for(int i=0; i<MAX; i++)
{
    cin>>arr[i];

    if(arr[i] > *mayor)
    {
        mayor = &arr[i];
    }

    if(arr[i] < *menor)
    {
       menor = &arr[i];
    }
}

cout << *menor << ' ' << *mayor;

This solution uses pointers instead of variables. Thus, menor does not store a value of its own, but will point to the smallest element (the same for mayor ).

If using pointers is not a possibility, then you should pay special attention to the initialization of menor and mayor . Pulling the standard the best is to use std::numeric_limits :

int menor = std::numeric_limits<int>::max();
int mayor = std::numeric_limits<int>::min();

What we do here is initialize menor with the largest number that can be stored in a int , in this way any number that the user enters will be equal or smaller and, consequently, the algorithm will be able to detect it as smaller. In mayor , instead, we store the smallest number possible, so that any number that the user enters is equal or larger.