Mathematics
rpearson345
10

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)

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jackie1977

[latex]f(x)=\ln x\implies f'(x)=\dfrac1x[/latex] [latex]g(x)=x^2-1\implies g'(x)=2x[/latex] You have [latex]2x^2>1[/latex] for all [latex]x\ge1[/latex], and dividing by [latex]x[/latex] gives [latex]2x>\dfrac1x[/latex], or [latex]g'(x)>f'(x)[/latex]. By the racetrack principle, then, it follows that [latex]g(x)>f(x)[/latex].

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