Question

y=e^x+sin(x)

How can you show that this equation has real roots? (Maybe with rolle?)

Answer 1

Roots occur when y=0, the equation transforms to e^x+sin(x)=0

Answer 2

Yes, but how can I proove that there are x's which solve this solution? :)

Answer 3

*this equation

Answer 4

Find y when x=-pi/2 and when x=0 :)

Answer 5

y changes sign between x=-pi/2 and x=0, therefore a root must exist in (-pi/2,0) because this function is continuous

Answer 6

And this function is continuous because e^x and sin(x) are both continuous

Answer 7

Because if y changes sign in x=[-pi/2,0], y must pass through 0.

Answer 8

that gives me 1 root (with the mean value theorem), but there is an infinite number of roots, right? (because of sin)

Answer 9

Yes, but this is out of the question's scope :)