Question

maximize ( b = 5xy^{2} ), where ( x ) and ( y ) are positive numbers such that ( x + y^{2}=12 ).
the maximum value of ( b ) is
(simplify your answer. type an exact answer, using radicals as needed)

Explanation:

Step1: Express \(x\) in terms of \(y\)

From \(x + y^{2}=12\), we get \(x = 12 - y^{2}\).

Step2: Substitute \(x\) into \(B\)

Substitute \(x = 12 - y^{2}\) into \(B = 5xy^{2}\), then \(B(y)=5(12 - y^{2})y^{2}=60y^{2}-5y^{4}\).

Step3: Find the derivative of \(B(y)\)

Using the power rule \((x^{n})^\prime=nx^{n - 1}\), \(B^\prime(y)=(60y^{2}-5y^{4})^\prime=120y-20y^{3}=20y(6 - y^{2})\).

Step4: Find the critical points

Set \(B^\prime(y)=0\). Since \(y>0\), \(20y(6 - y^{2}) = 0\) gives \(y^{2}=6\) (because \(y>0\), \(y=\sqrt{6}\)).

Step5: Find the second - derivative of \(B(y)\)

\(B^{\prime\prime}(y)=(120y - 20y^{3})^\prime=120-60y^{2}\).
When \(y^{2}=6\), \(B^{\prime\prime}(\sqrt{6})=120-60\times6=120 - 360=- 240<0\). So \(B(y)\) has a maximum at \(y^{2}=6\).

Step6: Find the value of \(x\)

When \(y^{2}=6\), from \(x = 12 - y^{2}\), we have \(x=12 - 6 = 6\).

Step7: Calculate the maximum value of \(B\)

Substitute \(x = 6\) and \(y^{2}=6\) into \(B = 5xy^{2}\), \(B=5\times6\times6 = 180\).

Answer:

\(180\)