2004 Q9 (c) (iii) (1 Viewer)

[meLoncoLLie]

f(x) = (lnx) / x, for x>0

maximum point at x = e

(iii) Use the fact that the maximum value of f(x) occurs at x = e
to deduce that e^x >= x^e for all x>0


I've looked at the solutions but still don't understand how to do it. Help anyone?

 

[acmilan]

when x = e, f(e) = lne/e = 1/e

Now the max value of f(x) is 1/e,

ie
lnx/x <= 1/e
elnx/x <= 1
elnx <= x
lnx^e <= x
x^e <= e^x
e^x >= x^e

 

[currysauce]

f(x) = (lnx) / x, for x>0

maximum point at x = e

(iii) Use the fact that the maximum value of f(x) occurs at x = e
to deduce that e^x >= x^e for all x>0


I've looked at the solutions but still don't understand how to do it. Help anyone?

ok if e is where the max is, then every other solution must be under it

so consider what f(e) = ln e / e = 1/e therefore the max point is (e,1/e)

therefore it follows all points are under this point

ln x / x < 1/e

then work through

 

[meLoncoLLie]

Thanks guys!