What number exceeds its square by the maximum amount
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What number exceeds its square by the maximum amount
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If we were to graph the number it would be:
y = x
If we were to graph the square it would be:
y = x²
The difference would be:
f(x) = x – x²
You want to maximize this difference, so take the
derivative:
f'(x) = 1 – 2x
Then set it to zero:
0 = 1 – 2x
Add 2x to both sides:
2x = 1
Divide both sides by 2:
x = ½
Answer: ½ is the number that most exceeds its square.