A lvl H2 Maths: Differentiation

The equation of a curve is given by lg X / X, x>0. Given that the only stationary point of the curve is a maximum, deduce without calculating any numerical values, that e^π > π^e

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Answer:

Use chain rule to differentiate lg x / x and set it to zero:

d/dx lg x / x = d/dx (1/x)(ln x)(1 / ln 10)
= (1 / ln 10) { 1 / x² - (ln x)(1/x²) = 0

1 - ln x = 0
x = e

So, x = e is the maximum point

thus,
lg e / e is the max value of the curve. Any other value is smaller (e.g. x=π).

lg e / e > lg π / π
π * lg e > e * lg π
lg e^π > lg π^e

Thus, we get e^π > π^e (solved)

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