Select Page

# 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
Medium
##### Suggested Book
Mathematical Circles

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 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

## Problem on Series | SMO, 2009 | Problem No. 25

Try this beautiful problem from Singapore Mathematics Olympiad, SMO, 2009 based on Problem on Series. You may use sequential hints to solve the problem.

## Area of The Region | AMC-8, 2017 | Problem 25

Try this beautiful problem from Geometry: The area of the region, AMC-8, 2017. You may use sequential hints to solve the problem.

## Area of the figure | AMC-8, 2014 | Problem 20

Try this beautiful problem from Geometry:Area inside the rectangle but outside all three circles.AMC-8, 2014. You may use sequential hints to solve the problem

## Squares and Triangles | AIME I, 2008 | Question 2

Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 2008 based on Squares and Triangles.

## Percentage Problem | AIME I, 2008 | Question 1

Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 2008 based on Percentage. you may use sequential hints.

## Smallest Positive Integer | PRMO 2019 | Question 14

Try this beautiful problem from the Pre-RMO, 2019 based on Smallest Positive Integer. You may use sequential hints to solve the problem.

## Complex Numbers and Triangles | AIME I, 2012 | Question 14

Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 2012 based on Complex Numbers and Triangles.

## Triangles and Internal bisectors | PRMO 2019 | Question 10

Try this beautiful problem from the Pre-RMO, 2019 based on Triangles and Internal bisectors. You may use sequential hints to solve the problem.

## Angles in a circle | PRMO-2018 | Problem 80

Try this beautiful problem from PRMO, 2018 based on Angles in a circle. You may use sequential hints to solve the problem.

## Circles and Triangles | AIME I, 2012 | Question 13

Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 2012 based on Circles and triangles.