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
1 answer:
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 = = 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
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