Problem 10 - Calculus Early Transcendentals

Question

What two nonnegative real numbers \(a\) and \(b\) whose sum is 23 maximize \(a^{2}+b^{2} ?\) Minimize \(a^{2}+b^{2} ?\)

Step-by-Step Solution

Answer

Answer: The minimum value of \(a^{2}+b^{2}\) is reached when \(a=11.5\) and \(b=11.5\). There is no maximum value for this expression under the given conditions.

Step 1: Express b in terms of a

Using the given constraint, \(a+b=23\), rearrange the equation to get \(b=23-a\). Now we have expressed \(b\) in terms of \(a\) and we can substitute this into the expression we need to optimize.

Step 2: Substitute b in the expression to optimize

We need to optimize the expression \(a^{2}+b^{2}\). We know that \(b=23-a\). Substitute this into the expression: \((a^{2})+((23-a)^{2})\) Expand this expression: \(a^{2}+(23^{2}-2\cdot23\cdot a+a^{2})\) Simplify: \(2a^{2}-46a+23^{2}\) Now we have a quadratic function in terms of \(a\).

Step 3: Find the vertex of the parabolic function

The vertex of the parabolic function can be found using the formula \(h=\frac{-b}{2a}\), where \(h\) is the \(a\)-coordinate of the vertex and \(a\) and \(b\) are the coefficients in the quadratic function of the form \(f(a) = 2a^{2}-46a+23^{2}\). In this case, \(a=2\) and \(b=-46\). Thus, \(h=\frac{-(-46)}{2\cdot2}=23/2=11.5\).

Step 4: Calculate b when a is 11.5

We found that the maximum or minimum value happens when \(a=11.5\). We now need to find the corresponding value of \(b\). Using the equation derived in Step 1 (\(b=23-a\)), we can calculate \(b\): \(b=23-11.5=11.5\)

Step 5: Determine whether the vertex represents a maximum or minimum

The quadratic function we found, \(f(a) = 2a^{2}-46a+23^{2}\), has a positive leading coefficient (\(2\)). Therefore, the parabola opens upwards, and the vertex represents a minimum. Thus, the minimum value of \(a^{2}+b^{2}\) occurs when \(a=11.5\) and \(b=11.5\). To find the maximum value, we note that there's no upper bound for the expression \(a^{2}+b^{2}\) given the constraint \(a+b=23\). Therefore, there is no maximum for this expression under the given conditions.

Step 6: Answer

The minimum value of \(a^{2}+b^{2}\) is reached when \(a=11.5\) and \(b=11.5\). There is no maximum value for this expression under the given conditions.