Let $a,$ $b,$ $c,$ $d$ be real numbers such that
\[a^2 + b^2 + c^2 + d^2 = 4.\]Find the maximum value of $a^3 + b^3 + c^3 + d^3.$
From the equation $a^2 + b^2 + c^2 + d^2 = 4,$ $a^2 \le 4,$ so $a \le 2,$ or $2 - a \ge 0.$  Then
\[(2 - a) a^2 \ge 0,\]so $a^3 \le 2a^2.$  Similarly, $b^3 \le 2b^2,$ $c^3 \le 2c^2,$ and $d^3 \le 2d^2.$  Adding all these inequalities, we get
\[a^3 + b^3 + c^3 + d^3 \le 2(a^2 + b^2 + c^2 + d^2) = 8.\]Equality occurs when $a = 2$ and $b = c = d = 0,$ so the maximum value is $\boxed{8}.$
8
Level 3
Intermediate Algebra
Problem: Let $a,$ $b,$ $c,$ $d$ be real numbers such that
\[a^2 + b^2 + c^2 + d^2 = 4.\]Find the maximum value of $a^3 + b^3 + c^3 + d^3.$
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To find the maximum value of \( a^3 + b^3 + c^3 + d^3 \) under the constraint \( a^2 + b^2 + c^2 + d^2 = 4 \), we can apply the method of Lagrange multipliers or utilize inequalities such as the Cauchy-Schwarz inequality.

First, we recognize that the expression \( a^3 + b^3 + c^3 + d^3 \) can be related to the squares of the variables. By the Cauchy-Schwarz inequality, we have:

\[
(a^2 + b^2 + c^2 + d^2)(a^4 + b^4 + c^4 + d^4) \geq (a^3 + b^3 + c^3 + d^3)^2.
\]

Substituting the constraint \( a^2 + b^2 + c^2 + d^2 = 4 \) into the inequality gives:

\[
4(a^4 + b^4 + c^4 + d^4) \geq (a^3 + b^3 + c^3 + d^3)^2.
\]

To maximize \( a^3 + b^3 + c^3 + d^3 \), we need to minimize \( a^4 + b^4 + c^4 + d^4 \). By the Power Mean inequality, we know that:

\[
\sqrt{\frac{a^4 + b^4 + c^4 + d^4}{4}} \geq \sqrt[4]{\frac{a^2 + b^2 + c^2 + d^2}{4}}.
\]

This implies:

\[
a^4 + b^4 + c^4 + d^4 \geq \frac{(a^2 + b^2 + c^2 + d^2)^2}{4} = \frac{16}{4} = 4.
\]

Thus, we have:

\[
4(a^4 + b^4 + c^4 + d^4) \geq 16.
\]

This leads to:

\[
(a^3 + b^3 + c^3 + d^3)^2 \leq 16,
\]

which implies:

\[
a^3 + b^3 + c^3 + d^3 \leq 4.
\]

However, we need to check if we can achieve a higher value. By setting \( a = 2 \) and \( b = c = d = 0 \), we find:

\[
a^2 + b^2 + c^2 + d^2 = 2^2 + 0 + 0 + 0 = 4,
\]

and

\[
a^3 + b^3 + c^3 + d^3 = 2^3 + 0 + 0 + 0 = 8.
\]

This configuration satisfies the constraint and yields \( 8 \) for \( a^3 + b^3 + c^3 + d^3 \). Therefore, the maximum value of \( a^3 + b^3 + c^3 + d^3 \) is indeed \( 8 \).<|END_OF_LATENT|> The final answer is $8$.