I am teaching a student in a course without partial differentiation so(adsbygoogle = window.adsbygoogle || []).push({});

I can only think of the following method

let b be a constant and set y=b so

xb= exp(x-1) +blnb

means b= exp(x-1) which means x = lnb+1

now d(xb)/dx= b

and d(exp(x-1)+blnb)/dx = exp(x-1)

so if x > lnb+1

xb> exp(x-1) + blnb

since the Right hand side has a higher derivative

so x > lnb+1 Implies xb> exp(x-1) + blnb

so

x > lny +1 implies xy< exp(x-1) + ylny

Now replace x with b

by = exp(b-1) + ylny

means

y=exp(b-1)

now d(by)/dx= b

and d( exp(b-1) + ylny)/dx = 1+ lny

so if lny+ 1 > b

by < exp(b-1) + ylny

since the derivative of the right hand is larger

so lny+1 >x implies xy < exp(x-1)+ylny

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# How to prove xy <= exp(x-1) +ylny using differentiation

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