Question

if a and b are positive numbers, find the maximum value of ( f(x)=x^{a}(5 - x)^{b} ) on the interval ( 0leq xleq5 ).

Explanation:

Step1: Find the derivative of \(f(x)\)

Use the product rule \((uv)^\prime = u^\prime v+uv^\prime\), where \(u = x^{a}\) and \(v=(5 - x)^{b}\).
\(u^\prime=ax^{a - 1}\), \(v^\prime=-b(5 - x)^{b - 1}\)
\(f^\prime(x)=ax^{a-1}(5 - x)^{b}-bx^{a}(5 - x)^{b - 1}=x^{a - 1}(5 - x)^{b - 1}[a(5 - x)-bx]\)
\(=x^{a - 1}(5 - x)^{b - 1}(5a-(a + b)x)\)

Step2: Find the critical points

Set \(f^\prime(x)=0\).
Since \(x\in[0,5]\), \(x^{a - 1}(5 - x)^{b - 1}\geq0\) for \(x\in[0,5]\).
Set \(5a-(a + b)x = 0\), then \(x=\frac{5a}{a + b}\)
Also, the endpoints \(x = 0\) and \(x = 5\)
\(f(0)=0^{a}(5-0)^{b}=0\)
\(f(5)=5^{a}(5 - 5)^{b}=0\)

Step3: Evaluate \(f(x)\) at the critical point

\(f(\frac{5a}{a + b})=(\frac{5a}{a + b})^{a}(5-\frac{5a}{a + b})^{b}\)
\(=(\frac{5a}{a + b})^{a}(\frac{5b}{a + b})^{b}=\frac{5^{a + b}a^{a}b^{b}}{(a + b)^{a + b}}\)

Answer:

\(\frac{5^{a + b}a^{a}b^{b}}{(a + b)^{a + b}}\)