Question

The number that exceeds its square by the greatest amount is _______________.

Answer 1

Let the number be x.

The square of the number is x².

Let f(x) = x − x². Now, we need to find the value of x for which f(x) is maximum.

f(x) = x − x²

Differentiating both sides with respect to x, we get

$f\text{'}\left(x\right)=1-2x$

For maxima or minima,

$f\text{'}\left(x\right)=0$

$\Rightarrow 1-2x=0$

$\Rightarrow x=\frac{1}{2}$

Now,

$f\text{'}\text{'}\left(x\right)=-2<0$

So, $x=\frac{1}{2}$ is the point of local maximum of f(x). Therefore, f(x) is maximum when $x=\frac{1}{2}$.

Thus, the number that exceeds its square by the greatest amount is $x=\frac{1}{2}$.


The number that exceeds its square by the greatest amount is $\underline{\frac{1}{2}}$.