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Question
Find the absolute maximum value and the absolute minimum value of the following function in the given interval:
f (x) = (x −1)2 + 3, x ∈[−3, 1]
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Solution
Given function f(x) = (x - 1)2 + 3 in the interval [-3, 1]
∴ f'(x) = 2(x - 1)
For critical points, let f' (x) = 0
If f'(x) = 0, then 2(x - 1) = 0,
⇒ x = 1 ∈ [-3, 1]
At, x = 1 f(1) = (1 - 1)2 + 3
= 0 + 3
= 3
At, x = -3 f(-3)
= (-3, -1)2 + 3
= 16 + 3 = 19
∴ Absolute maximum value of f(x) 19 at x = -3
Absolute minimum value of f(x) = 3 at x = 1.
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