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Maths Differentiation and Applications of Derivatives General Numeric Response

Published by: CGP EDU Academic Team

Published on: August 13, 2026

Find the minimum value of f(x) = 8 x + 8 – x – 4(4 x + 4 – x ), ∀ x ∈ R

Text Solution

Verified by Experts

The correct answer is:

(10)

Let u = 2 x + 2 –x

u 3 = 8 x + 8 –x + 3(2 x ) (2 –x ) (2 x + 2 –x )

⇒ u 3 – 3u = 8 x + 8 –x

also, 4 x + 4 –x = u 2 – 2

⇒ f(x) = u 3 – 3u – 4 (u 2 – 2) = u 3 – 4u 2 – 3u + 8

Let g(u) = u 3 – 4u 2 – 3u + 8 ; u > 2

g’(u) = 3u 2 – 8u – 3 = (3u + 1) (u – 3)

putting g’(u) = 0 ; we get u = 3

g’’ (u) = 6u – 8 ⇒ g’’(3) = 1 > 0

⇒ u = 3 is point of minima

⇒ g(3) = 27 – 36 – 9 + 8 = –10

⇒ minimum f(x) = – 10

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