Find-the-relation-between-m-and-n-for-which-the-following-holds-d-y-d-x-x-n-d-x-d-y-y-m-1-

Question Number 207450 by York12 last updated on 15/May/24

Find the relation between m and n for which the following holds

\frac{d (y)}{d (x)} ∣_{x = n} = {(\frac{d (x)}{d (y)} ∣_{y = m})}^{- 1}

Answered by mr W last updated on 15/May/24

Commented by mr W last updated on 15/May/24

\frac{d y}{d x} ∣_{x = n} = \tan θ

\frac{d x}{d y} ∣_{y = m} = \tan ϕ

{(\frac{d x}{d y} ∣_{y = m})}^{- 1} = \frac{1}{\tan ϕ} = \tan φ

\frac{d y}{d x} ∣_{x = n} = {(\frac{d x}{d y} ∣_{y = m})}^{- 1}

\Rightarrow \tan θ = \tan φ

\Rightarrow θ = φ

i f y = f (x) i s a s t r i c t l y m o n o t o n i c

f u n c t i o n, t h e n P a n d Q m u s t b e

t h e s a m e p o i n t, i . e . m = f (n) .

i f y = f (x) i s n o t a s t r i c t l y m o n o t o n i c

f u n c t i o n, t h e n t h e r e l a t i o n b e t w e e n

n u n d m i s n o t u n i q u e .

Commented by York12 last updated on 15/May/24

thank you sir

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