Question

Calculus help??????

the functions

f(x)= ln(x) and g(x)= x^2-1

they are both 0 when x=1

apply the Race Track Principle to explain which of the following inequalities is true for x is (greater than or equal to) 1:

f(x) < g(x)
or
g(x) < f(x)

Answer 1

`f(x)=\ln x\implies f'(x)=\frac1x`

`g(x)=x^2-1\implies g'(x)=2x`

You have
`2x^2>1` for all
`x\ge1`, and dividing by
`x` gives
`2x>\frac1x`, or
`g'(x)>f'(x)`. By the racetrack principle, then, it follows that
`g(x)>f(x)`.