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.

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