For a twice-differentiable function with on an open interval containing

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

Problem

For a twice-differentiable function

f

with

f''(x)<0

on an open interval containing

c

, which statement about the linearization

L(x)

f

x=c

is true for

x\neq c

sufficiently close to

c

? (a)

L(x)

is less than

f(x)

for all such

x

(the linearization is an under–estimate). (b)

L(x)

is greater than

f(x)

for all such

x

(the linearization is an over–estimate). (c)

L(x)

is greater than

f(x)

for

x>c

and less than

f(x)

for

x<c

. (sign depends on which side of

c

) (d)

L(x)=f(x)

only at

x=c

and is less than

f(x)

for every other

x

near

c