Find the values of x if f x - x x 2 -1 is i an increasing functionii a decreasing function

f( x ) = x x ^ 2 +1 f^ ' ( x ) =( x ^ 2 +1 ) ( 1 ) x( 2x ) #160; ( x ^ 2 +1 ) #160; ^ 2 f'(x)= 1 x ^ 2 ( x ^ 2 +1 ) #160; ^ 2 f' ...

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Monotonically Increasing Functions

Find the valu...

Question

Find the values of x if $f (x) = \frac{x}{x^{2} + 1}$ is
i) an increasing function
ii) a decreasing function

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Solution

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f (x) = x^{3} + λ x^{2} - λ x + 1

x = 0

x = 1

λ

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

x

x

f (x) = x^{3} + \frac{1}{x^{3}} (x \neq 0)

R

a

f (x) = x^{2} + a x + 1

[1, 2]

S

a

f (x) = x^{2} + a x + 1

[1, 2]

| R - S |

f : A \to B

f

f : R^{+} \to R^{+}

f (x) = x^{2}

f^{- 1} (x) = \sqrt{x}

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