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The maximum value of [x (x - 1) + 1]1/3, 0 ≤ x ≤ 1 is
(A) (1/3)1/3 (B) 1/2 (C) 1 (D) 0
Solution:
Maxima and minima are known as the extrema of a function
Maxima and minima are the maximum or the minimum value of a function within the given set of ranges.
Let f (x) = [x (x - 1) + 1]1/3
Therefore,
On differentiating wrt x, we get
f' (x) = (2x - 1)/[x (x - 1) + 1]2/3
Now,
f' (x) = 0
⇒ x = 1/2
Then, we evaluate the value of f at critical point at x = 1/2 and at the end points of the interval [0, 1]
i.e., at x = 0 and x = 1.
f (0) = [0 (0 - 1) + 1]1/3
f (1) = [1(1 - 1) + 1]1/3
f (1/2) = [1/2(- 1/2) + 1/2]1/3
= (3/4)1/3
Hence, we can conclude that the maximum value of f in the interval [0, 1] is 1.
Thus, the correct option is C
NCERT Solutions Class 12 Maths - Chapter 6 Exercise 6.5 Question 29
The maximum value of [x (x - 1) + 1]1/3, 0 ≤ x ≤ 1 is (A) (1/3)1/3 (B) 1/2 (C) 1 (D) 0
Summary:
The maximum value of [x (x - 1) + 1]1/3, 0 ≤ x ≤ 1 is 1. Thus, the correct option is C
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