A Mysterious Recursion

Since, x can have only integer values & x+10 is strictly increasing.

Therefore, (x+10) is max at x=9 i.e. f(9)=19.

Now,
At x=10; f(10)=f(10-5)= f(5)= 5+10=15;
At x=11; f(11)=f(11-5)=f(6)=16;
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At x=14; f(14)= f(14-5)=19;
& At x=15; f(15)= f(15-5)=f(10)= f(10-5)= f(5)= 15
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And thus the function periodically repeats itself and the max value achieved during this repetition is 19.

Since 9 is the largest value you can get in the x < 10 Then 9 + 10 = 19 19 is the answer

9+10=19

for values greater than 10 function will go back like f(12) will be f(7) likewise so maximum value will be at 9 i. e. 19

In this function, we can see that if x>=10, then f(x)=f(x-5) and thus we can see that if we give any value of x>=10, it finally gets reduced to a form of f(x) with x<10. And so, the values of f(x) only lies between 0<x<10. We see that we get highest value of f(x) in $f(9)=9+10=\boxed{19}$

f(x)=x+10 when x<10, put x value from 1 to 9. the last value is 9 so f(x)=9+10=19.

The recursion implies that all values of x greater than or equal to 10 will be reduced until it becomes less than 10. And for a value of x less than 10, the function has the greatest value at x = 9 in 19.

$f(9)$ should be greatest, as $x=10 to 14$ ends up $f(x)=f(5) to f(9)$

This is a periodic relation. So, highest value, $f(9)=10+9=19$