For a certain two-part journey

MATHEMATICS · Single correct · CBSE · Mathematics · Applications of Derivatives · Applications of Derivatives

Problem

For a certain two-part journey, the total travel time is modeled by a differentiable function

T(x)

, where

x

is a controllable distance along the first part of the path and

a<x<b

. Suppose

T'(c)=0

at a single point

x=c

and

T'(x)<0

for

a<x<c

while

T'(x)>0

for

c<x<b

. Which statement best justifies that the path corresponding to

x=c

minimizes the travel time? (a) Because

T''(c)<0

, the function is concave down at

x=c

, so this point must give the least travel time. (b) The change in

T'(x)

from negative to positive at

x=c

shows a local minimum by the First Derivative Test, so the travel time is minimized there. (c) A sign change of

T'(x)

from positive to negative at

x=c

guarantees a local minimum, thus

x=c

yields the least time. (d) Since

T'(x)=0

only at

x=c

, the minimum and maximum must both occur there; therefore the travel time is smallest at

x=c