91Ó°ÊÓ

Since we know that the sum of a and b is 23, we can write \(b\) in terms of \(a\) as follows: \(b = 23 - a\).

Now, we will substitute the value of \(b\) found in step 1 into the expression \(a^2 + b^2\): \(a^2 + (23 - a)^2 = a^2 + (529 - 46a + a^2)\). This simplifies the expression to \(2a^2 - 46a + 529\).

Now, we will find the derivative of the expression \(2a^2 - 46a + 529\) with respect to \(a\): \(\frac{d}{da}(2a^2 - 46a + 529) = 4a - 46\).

To find the critical points, we need to set the derivative equal to 0 and solve for \(a\): \(4a - 46 = 0\). Solving for \(a\), we get \(a = \frac{46}{4} = 11.5\).

Now, we will use the value of \(a\) found in step 4 to determine the value of \(b\): \(b = 23 - a = 23 - 11.5 = 11.5\).

We only have one critical point, \((11.5, 11.5)\), so this must correspond to either the maximum or minimum value of \(a^2 + b^2\). Since \(a\) and \(b\) are nonnegative real numbers, the lowest possible value for \(a^2 + b^2\) is 0, which occurs when \(a = b = 0\). However, this doesn't satisfy the constraint that \(a + b = 23\). Therefore, the critical point \((11.5,11.5)\) must correspond to the maximum value of \(a^2 + b^2\). The maximum value is \(11.5^2 + 11.5^2 = 265.25\). As for the minimum value, we know it cannot be 0. So, we need to consider the boundary of our domain. Since \(a\) and \(b\) are nonnegative, the minimum value will occur when either \(a\) or \(b\) is 0. We have two possibilities: \((a,b) = (23,0)\) or \((a,b) = (0,23)\). In both cases, \(a^2 + b^2 = 23^2 = 529\). So, the minimum value of \(a^2 + b^2\) is 529.

The two nonnegative real numbers \(a\) and \(b\) whose sum is 23 that maximize \(a^2 + b^2\) are \(a = 11.5\) and \(b = 11.5\), with a maximum value of 265.25. The two nonnegative real numbers \(a\) and \(b\) that minimize \(a^2 + b^2\) are either \(a = 23, b = 0\) or \(a = 0, b = 23\), with a minimum value of 529.