Prove that this equation has at least one real root

[verty]

I gave that hint about adjacent roots but I should have thought more about the question first, because it (the question as well as my hint) was a little misleading. Dick (as usual, I'm pleased to say) led us directly to the best way to look at it, a geometric argument.

 

[haruspex]

I gave that hint about adjacent roots but I should have thought more about the question first, because it (the question as well as my hint) was a little misleading. usual, I'm pleased to say) led us directly to the best way to look at it, a geometric argument.

Thanks for clearing that up, verty. (I still like my e^λx method.)

 

[Dick]

Thanks for clearing that up, verty. (I still like my e^λx method.)

I like it too. It's elegant and does use pure Rolle's theorem, not the MVT. But I like the MVT approach because I think it's easier to discover graphically, without the extra bit of cleverness. Tough call really.