Let be twice–differentiable with for all in an open interval containing

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

Problem

Let

f

be twice–differentiable with

f''(x)>0

for all

x

in an open interval containing

a

. For

x>a

that is still in this interval, which statement describes the relationship between the tangent-line approximation of

f(x)

x=a

and the actual value

f(x)

? (a) The approximation overestimates

f(x)

because the graph is concave up. (b) The approximation underestimates

f(x)

only when

x<a

. (c) The approximation underestimates

f(x)

because the graph is concave up. (d) If

f'(a)>0

, the approximation overestimates

f(x)

; if

f'(a)<0

, it underestimates.