Use app×
Join Bloom Tuition
One on One Online Tuition
JEE MAIN 2025 Foundation Course
NEET 2025 Foundation Course
CLASS 12 FOUNDATION COURSE
CLASS 10 FOUNDATION COURSE
CLASS 9 FOUNDATION COURSE
CLASS 8 FOUNDATION COURSE
+1 vote
6.2k views
in Limit, continuity and differentiability by (52.5k points)

Let f be a differentiable function on the open interval (a, b). Which of the following statements must be true?

I. f is continuous on the closed interval [a, b]

II. f is bounded on the open interval (a, b)

III. If a < a1< b1 < b, and f(a1) < 0 < f(b1), then there is a number c such that a1 < c < b, and f(c) = 0

(A) I and II only 

(B) I and III only 

(C) II and III only 

(D) only III

1 Answer

0 votes
by (54.8k points)
selected by
 
Best answer

Answer is (D) only III

I and II are false. The function f(x) = 1/x, 0 < x <1, is a counter example.

Statement III is true.

Apply the intermediate value theorem to f on the closed interval [a1, b1]

Welcome to Sarthaks eConnect: A unique platform where students can interact with teachers/experts/students to get solutions to their queries. Students (upto class 10+2) preparing for All Government Exams, CBSE Board Exam, ICSE Board Exam, State Board Exam, JEE (Mains+Advance) and NEET can ask questions from any subject and get quick answers by subject teachers/ experts/mentors/students.

...