 # 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.

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