If `m` is the slope of a tangent to the curve `e^y=1+x^2,` then (a)`|m|>1` (b) `m >1` (c)`m >=-1` (d) `|m|lt=1`

Options

A. `|m| gt 1 `
B. `m lt 1 `
C. `|m| lt 1 `
D. `|m| le 1 `

Answer

D

Explanation

We have,
` e^(y)=1+x^(2) `
` rArr e^(y) (dy)/(dx)=2x " " [" Differentiating w.r.t. x" ] `
` rArr (1+x^(2))(dy)/(dx)=2x " " [ because e^(y)=1+x^(2)] `
` rArr (dy)/(dx) = (2x)/(1+x^(2)) `
` rArr |m|= (2|x|)/(1+|x|^(2)) `
Now, A.M.` ge ` G.M.
` rArr (1+|x|^(2))/(2) ge sqrt(1xx |x|^(2)) `
` rArr (1+|x|^(2))/(2) ge |x| `
` rArr 1+|x|^(2) ge 2|x| `
` rArr 1 ge (2|x|)/(1+|x|^(2)) rArr 1 ge |m| rArr |m| le 1 `