What number exceeds its square by the greatest amount?

To find the number that exceeds its square by the greatest amount, we need to maximize the expression:

\[
f(x) = x - x^2
\]

### Step 1: Understanding the Function

The function \( f(x) = x - x^2 \) represents the difference between a number \( x \) and its square \( x^2 \). We want to find the value of \( x \) that maximizes this expression.

### Step 2: Find the Critical Points

To find the critical points, we first take the derivative of \( f(x) \) with respect to \( x \) and set it to zero:

\[
f'(x) = 1 - 2x
\]

Set \( f'(x) = 0 \):

\[
1 - 2x = 0
\]

\[
2x = 1
\]

\[
x = \frac{1}{2}
\]

### Step 3: Determine if It Is a Maximum

To determine if \( x = \frac{1}{2} \) is a maximum, we check the second derivative:

\[
f''(x) = -2
\]

Since \( f''(x) = -2 < 0 \), the function \( f(x) = x - x^2 \) has a maximum at \( x = \frac{1}{2} \).

### Step 4: Calculate the Maximum Value

Now, we substitute \( x = \frac{1}{2} \) back into the function to find the maximum value:

\[
f\left(\frac{1}{2}\right) = \frac{1}{2} - \left(\frac{1}{2}\right)^2
\]

\[
f\left(\frac{1}{2}\right) = \frac{1}{2} - \frac{1}{4}
\]

\[
f\left(\frac{1}{2}\right) = \frac{2}{4} - \frac{1}{4}
\]

\[
f\left(\frac{1}{2}\right) = \frac{1}{4}
\]

### Conclusion

The number that exceeds its square by the greatest amount is \( \frac{1}{2} \). The maximum amount by which it exceeds its square is **\(\frac{1}{4}\)**.

Would you like further details on any part of this solution, or have any questions?

### Related Questions

1. How do you find the maximum or minimum of a quadratic function?
2. What is the graphical interpretation of \( f(x) = x - x^2 \)?
3. How does the second derivative test determine concavity and maxima?
4. What happens if we consider only integer values for the number?
5. How would you solve similar problems involving other polynomial functions?

### Tip:
When maximizing or minimizing expressions, always consider using derivatives to find critical points and analyze their nature.

Discover how to find the number that exceeds its square by the greatest amount using calculus. Learn about maximizing quadratic functions and applying the second derivative test. Suitable for advanced high school students.

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