Consider the function [tex]f(x)=x^{1/3}[/tex], which has derivative [tex]f'(x)=\dfrac13x^{-2/3}[/tex].
The linear approximation of [tex]f(x)[/tex] for some value [tex]x[/tex] within a neighborhood of [tex]x=c[/tex] is given by
[tex]f(x)\approx f'(c)(x-c)+f(c)[/tex]
Let [tex]c=64[/tex]. Then [tex](63.97)^{1/3}[/tex] can be estimated to be
[tex]f(63.97)\approxf'(64)(63.97-64)+f(64)[/tex]
[tex]\sqrt[3]{63.97}\approx4-\dfrac{0.03}{48}=3.999375[/tex]
Since [tex]f'(x)>0[/tex] for [tex]x>0[/tex], it follows that [tex]f(x)[/tex] must be strictly increasing over that part of its domain, which means the linear approximation lies strictly above the function [tex]f(x)[/tex]. This means the estimated value is an overestimation.
Indeed, the actual value is closer to the number 3.999374902...
