Question

If a and b are positive numbers, find the maximum value of f(x)=xa(1−x)b, 0≤x≤1 Your answer may depend on a and b.maximum value =________.

Answer 1

Answer:

0.25ab

Step-by-step explanation:

Data provided in the question:

f(x) = xa(1−x)b, 0≤x≤1

or

f(x) = ab(x−x²)

for point of maxima and minima put f'(x) = 0

Thus,

f'(x) = ab(1 - 2x) = 0

or

1 - 2x = 0

or

x = $\frac{1}{2}$ = 0.5

Now,

to check the condition of maxima or minima

f''(x) = ab(0 - 2) = -2ab

since,

f''(x) < 0

therefore,

x = 0.5 is point of maxima

and the maximum value at x = 0.5 of the function is

f(0.5) = ab(0.5 - 0.5²)

= ab(0.25)

= 0.25ab