For a twice-differentiable function

MATHEMATICS · Multi-part · CBSE · Mathematics · Applications of Derivatives · Applications of Derivatives

Problem

f

, the local behavior of the graph near a point can be related to the sign of

f''

at that point. Answer the following questions in the context of this idea. (1) Explain, without specific calculations, how the sign of

f''(a)

influences whether the tangent line to

y=f(x)

x=a

lies above or below the graph of

f

for

x

near

a

. Your explanation should refer to the geometric meaning of concavity. (2) Let

f(x)=e^{x}

. Compute

f''(0)

and state its sign. (3) Using your result from part (b), discuss whether the tangent line to

y=f(x)

x=0

gives an overestimate or an underestimate of

f(x)

for

x>0

sufficiently close to

0

. Justify your conclusion thoroughly.