Application of DerivativeMedium
Question
On the interval [0, 1] the function x25(1 - x)75 takes its maximum value at the point
Options
A.0
B.1/4
C.1/2
D.1/3
Solution
Let f(x) = x25 (1 - x)75, x ∈ [0, 1]
⇒ f′(x) = 25x24 (1- x)75 - 75x25 (1 - x)74
25x24 (1 - x)74[(1 - x) - 3x]
= 25x24 (1 - x)74 (1 - 4x)
For maximum value of f(x) put f′(x) = 0
⇒ 25x24 (1 - x)74 (1 - 4x) = 0
⇒ x = 0, 1,
Also, at x = 0, y = 0
at x = 1, y = 0
and x = 1/ 4, y > 0
∴ f(x) attains maximum at x = 1/4
⇒ f′(x) = 25x24 (1- x)75 - 75x25 (1 - x)74
25x24 (1 - x)74[(1 - x) - 3x]
= 25x24 (1 - x)74 (1 - 4x)
For maximum value of f(x) put f′(x) = 0
⇒ 25x24 (1 - x)74 (1 - 4x) = 0
⇒ x = 0, 1,
Also, at x = 0, y = 0
at x = 1, y = 0
and x = 1/ 4, y > 0
∴ f(x) attains maximum at x = 1/4
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