Let f : R → R be such that for all x ∈ R 2 1 + x + 2 1 - x , f x and 3 x + 3 - x are in A.P., then the minimum value of f x is

f x = 2 1 - x + 2 1 + x + 3 x + 3 - x 2 ⇒ f x = 2 3 - x + 3 x + 3 x + 3 - x 2 ⇒ f x = 3 - x + 3 x + 3 x + 3 - x 2 Using A M ≥ G M f x ≥ 3

Mathematics - Sequence & Series Question with Solution | TestHub

MathematicsSequence & SeriesA.P.Medium2 minPYQ_2020

MathematicsMediumstatement

Let $f : R \to R$ be such that for all $x \in R (2^{1 + x} + 2^{1 - x}), f (x)$ and $(3^{x} + 3^{- x})$ are in A.P., then the minimum value of $f (x)$ is

Options:

Answer:

Solution:

$f (x) = (\frac{2^{1 - x} + 2^{1 + x} + 3^{x} + 3^{- x}}{2})$

$\Rightarrow f (x) = (\frac{2 (3^{- x} + 3^{x}) + (3^{x} + 3^{- x})}{2})$

$\Rightarrow f (x) = ((3^{- x} + 3^{x}) + (\frac{3^{x} + 3^{- x}}{2}))$

Using $A M \geq G M$
$f (x) \geq 3$

Stream:JEESubject:MathematicsTopic:Sequence & SeriesSubtopic:A.P.

⏱ 2mℹ️ Source: PYQ_2020

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