Understand the problem

Let f : [0, 2] → R be a continuous function such that \( \frac{1}{2} .\int^2_0 f(x)\,dx < f(2)\).
Then which of the following statements must be true?

Look at the knowledge graph.

Source of the problem
I.S.I. B.Stat. Entrance 2017, UGA Problem 20
Key Competency
Maximum and minimum property of  function 
Difficulty Level
Medium
Suggested Book
Mathematical Circles

Start with hints

Do you really need a hint? Try it first!

See that from the given result we have  \( \frac{1}{2}.\int^2_0 f(x)\,dx < f(2) \Rightarrow \int^2_0 f(x)\,dx < 2.f(2) \Rightarrow \int^2_0 f(x)\,dx < \int^2_0 f(2)\,dx \).
Now if f(2) is minimum then f(x)>f(2) for all x belong to [0,2] . Therefore, \(\int^2_0 f(x)\,dx > \int^2_0 f(2)\,dx \), which gives contradiction to the given result \(\int^2_0 f(x)\,dx < \int^2_0 f(2)\,dx \).    
From this we can’t say that f must be strictly increasing or f must attain a maximum value at x = 2.The only thing we can say that  f cannot have a minimum at x = 2.

Connected Program at Cheenta

Math Olympiad Program

Math Olympiad is the greatest and most challenging academic contest for school students. Brilliant school students from over 100 countries participate in it every year. Cheenta works with small groups of gifted students through an intense training program. It is a deeply personalized journey toward intellectual prowess and technical sophistication.

Similar Problems

Graph Coordinates | AMC 10A, 2015 | Question 12

Try this beautiful Problem on Graph Coordinates from co-ordinate geometry from AMC 10A, 2015. You may use sequential hints to solve the problem.

Digits of number | PRMO 2018 | Question 3

Try this beautiful problem from the Pre-RMO, 2018 based on Digits of number. You may use sequential hints to solve the problem.

Smallest value | PRMO 2018 | Question 15

Try this beautiful problem from the Pre-RMO, 2018 based on the Smallest value. You may use sequential hints to solve the problem.

Length and Triangle | AIME I, 1987 | Question 9

Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1987 based on Length and Triangle.

Positive Integer | PRMO-2017 | Question 1

Try this Integer Problem from Algebra from PRMO 2017, Question 1 You may use sequential hints to solve the problem.

Algebra and Positive Integer | AIME I, 1987 | Question 8

Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1987 based on Algebra and Positive Integer.

Distance and Spheres | AIME I, 1987 | Question 2

Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1987 based on Distance and Spheres.

Distance Time | AIME I, 2012 | Question 4

Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 2012 based on Distance Time. You may use sequential hints.

Arithmetic Mean | AIME I, 2015 | Question 12

Try this beautiful problem from the American Invitational Mathematics Examination, AIME, 2015 based on Arithmetic Mean. You may use sequential hints.

Algebraic Equation | AIME I, 2000 Question 7

Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 2000 based on Algebraic Equation.