A three-times differentiable function satisfies for all real and

MATHEMATICS · Single correct · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation

Problem

A three-times differentiable function

r

satisfies

r''(x)=x-2

for all real

x

and

r'''(x)>0

. Let

T(x)

be the tangent line to

r

x=2

. Which statement best describes whether

T(x)

overestimates or underestimates

r(x)

for values of

x

sufficiently close to

2

? (a)

T(x)

underestimates

r(x)

on both sides of

x=2

. (b)

T(x)

overestimates

r(x)

for

x<2

and underestimates

r(x)

for

x>2

. (c)

T(x)

underestimates

r(x)

for

x<2

and overestimates

r(x)

for

x>2

. (d)

T(x)

overestimates

r(x)

on both sides of

x=2