If a^2+b^2=8, what is the largest a+b could be, and what is the largest a*b could be, and prove it mathematically.

Given the equation \( a^2 + b^2 = 8 \), we need to find the maximum value of \( a + b \) and the maximum value of \( a \times b \). 

### Step 1: Maximizing \( a + b \)

We start by expressing \( b \) in terms of \( a \) and substituting into the equation \( a^2 + b^2 = 8 \).

Let \( b = x - a \). Then,

\[
a^2 + (x - a)^2 = 8
\]

Expanding and simplifying:

\[
a^2 + (x^2 - 2ax + a^2) = 8
\]

\[
2a^2 - 2ax + x^2 = 8
\]

This can be treated as a quadratic equation in \( a \):

\[
2a^2 - 2ax + (x^2 - 8) = 0
\]

The discriminant \( \Delta \) must be non-negative for \( a \) to have real values:

\[
\Delta = (-2x)^2 - 4 \times 2 \times (x^2 - 8)
\]

\[
\Delta = 4x^2 - 8(x^2 - 8) = -4x^2 + 64
\]

For \( \Delta \geq 0 \), we require:

\[
-4x^2 + 64 \geq 0
\]

\[
x^2 \leq 16
\]

So, \( -4 \leq x \leq 4 \).

To maximize \( a + b = x \), we find \( x \) within this range, and the maximum occurs at \( x = 4 \).

### Step 2: Maximizing \( ab \)

To maximize \( ab \), we use the identity:

\[
(a + b)^2 = a^2 + b^2 + 2ab
\]

Given \( a^2 + b^2 = 8 \), we substitute:

\[
(a + b)^2 = 8 + 2ab
\]

Let \( s = a + b \) and \( p = ab \). Then:

\[
s^2 = 8 + 2p
\]

Thus, the equation becomes:

\[
2p = s^2 - 8
\]

\[
p = \frac{s^2 - 8}{2}
\]

Since \( -4 \leq s \leq 4 \), to maximize \( p \), substitute \( s = 0 \) or \( s = \pm 4 \):

- If \( s = 4 \), then:

\[
p = \frac{4^2 - 8}{2} = \frac{16 - 8}{2} = 4
\]

- If \( s = 0 \), then:

\[
p = \frac{0^2 - 8}{2} = \frac{-8}{2} = -4
\]

The maximum value of \( ab \) is 4 when \( s = 4 \).

### Conclusion

- The **largest \( a + b \)** can be is **4**.
- The **largest \( ab \)** can be is **4**.

These maximums occur when \( a = 2 \) and \( b = 2 \) or vice versa.

Would you like more details or have any questions? Here are some related problems you might explore:

1. If \( a^2 + b^2 = 10 \), what are the maximum values of \( a + b \) and \( ab \)?
2. How would the solutions change if \( a^2 + b^2 \leq 8 \) instead of \( a^2 + b^2 = 8 \)?
3. What happens to the maximum \( ab \) if \( a \) and \( b \) are constrained to be integers?
4. Explore the case where \( a^2 + b^2 = k \) for any positive integer \( k \). How does \( k \) affect the maximum values?
5. If \( a^2 + b^2 = c^2 \) for some constant \( c \), how does this affect the solutions?

**Tip:** When maximizing functions involving sums and products, express the variables in terms of each other using identities and then apply known inequalities like the discriminant or the Cauchy-Schwarz inequality.

Explore how to maximize the sum and product of two variables, a and b, given the equation a^2 + b^2 = 8. Learn step-by-step solutions and mathematical proofs to find the maximum values. Suitable for advanced high school students.

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