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

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}})