Need help with 2 Qs (1 Viewer)

[Hikari Clover]

do i need to worry about those asymptotes?

 

[TesseracT22]

Ok i did the first question. The differentiation is easy, just change the base to base e and then quotient rule to differentiate.

h(x)= 1/ln10 (lnx/x)

h2(x)= 1/ln10 [(1-lnx)/x]

2nd part was a bit harder, there's some clues in the question such as "stationary point is max" so you let h2(x)=0.

you end up with x=e at the max turning point.
sub that back into h(x)

so H(e)= 1/ln10 (lne/e)
simplify to

H(e) = 1/eln10, now the answer has a pi in it so some how you have to get that in an expression using the fact that 1/eln10 is the largest y value on this graph.

So you can have an expression such as 1/eln10 > h(x) if x does not =e
simplify to 1/e > lnx/x now let x=pi because this inequality is tru for all x besides e.

Now apply e^ to both sides and the inequality still holds becuase e>0

so you get e^(1/e) > e^(ln(Pi)/Pi)
which is

e^(1/e) > [e^ln(pi)]^(1/pi)

e^(1/e) > Pi^(1/pi)

raise both sides by powers of pi and e inequality holds because e, pi>0

therefore: e^pi > pi^e

 

[aalex]

Hope it helps.
Good luck!!!