Exponential Question (Harder 3u) (1 Viewer)

[K4M1N3]

This is a question i found in my half yearly 4u exam, just wanted to see how people go about answering it.

For the equation: <a href="http://www.codecogs.com/eqnedit.php?latex=e^x@plus;e^y=5" target="_blank"><img src="http://latex.codecogs.com/gif.latex?e^x+e^y=5" title="e^x+e^y=5" /></a>

i) Prove that it is monotonous decreasing
ii) Prove that its concavity is downward
iii) Sketch the curve

 

[ohexploitable]

I haven't studied Harder 3U but...

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First two parts are straightforward, just differentiate implicitly etc.

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For the graph, first you find the intercepts,



If you rearrange the equation you get,



The domain is restricted,



For any horizontal asymptotes,



So there's a vertical asymptote at x=log5 and a horizontal one at y=log5,

Using parts (i) and (ii), the shape should be fairly straightforward.