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F ( X )=4 X 2 -4 X + 4 on R . - Mathematics

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Question

f(x) = 4x2 + 4 on R .

Solution

Given: f(x) = 4x2 − 4x + 4

\[\Rightarrow\] f(x) = (4x2 − 4x + 1)+3

\[\Rightarrow\] f(x) = (2x − 1)2 + 3

Now,
(2x − 1)\[\geq\] 0 for all x \[\in\] R \[\in\]

\[\Rightarrow\] f(x) = (2x − 1)2 + 3 \[\geq\] 3 for all x \[\in\] R

\[\Rightarrow\] f(x) \[\geq\] 3 for all x \[\in\] R

The minimum value of f is attained when (x − 1) = 0.
(2x − 1) = 0
⇒ x = \[\frac{1}{2}\]

Thus, the minimum value of f (x) at x = \[\frac{1}{2}\] is 3.

Since f(x) can be enlarged, the maximum value does not exist, which is evident in the graph also.

Hence, function f does not have a maximum value .

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APPEARS IN

 RD Sharma Solution for Mathematics for Class 12 (Set of 2 Volume) (2018 (Latest))
Chapter 18: Maxima and Minima
Ex. 18.1 | Q: 1 | Page no. 7
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F ( X )=4 X 2 -4 X + 4 on R . Concept: Graph of Maxima and Minima.
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