Question

Prove that the curve `y=e^(|x|)` cannot have a unique tangent line at the point `x = 0.` Find the angle between the one-sided tangents to the curve at the point `x = 0.`
A. `(pi)/(4) `
B. `(pi)/(6) `
C. `(pi)/(2) `
D. `(pi)/(3) `

Answer 1

Correct Answer - C
We have,
`y=e^(|x|)={(e^(-x)"," , x lt0),(e^(x) "," , x ge 0):} rArr (dy)/(dx)={(-e^(-x)"," , x lt0),(e^(x) "," , x gt 0):} `
The slopes of the tangents at x = 0 to the curves ` y=e^(-x), x lt 0 ` and `y= e^(x), x ge 0 " are " m_(1)=-1 " and " m_(2)=1 ` respectively.
Clearly , ` m_(1)m_(2)=-1 `
Hence , the required angle is a right angle.