Let . Which of the following statements best justifies that has at least one real…

MATHEMATICS · Single correct · AP · Calculus AB · Limits and Continuity · Limits and Continuity

Problem

Let

g(x)=\ln x-2\sin x

. Which of the following statements best justifies that

g

has at least one real zero in the interval

(1,3)

? (a) Because

g

is differentiable on

(1,3)

and

g(3)-g(1)\neq0

, the Mean Value Theorem ensures some

c \in (1,3)

with

g(c)=0

. (b) Since

\ln x

and

\sin x

are periodic functions,

g

must take on every value between its maximum and minimum on

(1,3)

, including

0

. (c)

g

is continuous on

[1,3]

and

g(1)<0

while

g(3)>0

; therefore, by the Intermediate Value Theorem there exists

c \in (1,3)

with

g(c)=0

. (d) Because

\ln 1=0

, we have

g(1)=0

; continuity then forces

g

to remain

0

in some interval to the right of

1