Mathematics

Problem

Consider

f(x)=x^{3}

and

g(x)=7x^{2}

for

x>0

. On which interval does

f

increase faster than

g

? a)

(7,\,\infty)

(\dfrac{14}{3},\,\infty)

(0,\,\dfrac{14}{3})

d) Only at

x=\dfrac{14}{3}

Intuitive explanation

The key is comparing the rates of change of these functions using their derivatives. Since $f'(x) = 3x^2$ and $g'(x) = 14x$ , you're asking where $f$ has a steeper slope than $g$ —that is, where $f'(x) > g'(x)$ . This inequality becomes a simple comparison of two expressions in $x$ , which you can solve algebraically to find the boundary points.

Intuitive explanation

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