No computational aids required!

Calculus Level 5

Let $f : [0,1] \to \mathbb{R}$ be a function defined as follows: $\large f(x) = 10 x^{\sin x + \cos x}$

Enter the value of the following expression:

$\left \lfloor \int_0^1 f(x) \, dx\right \rfloor$

Details and Assumptions:

$\lfloor \cdot \rfloor$ denotes the floor function .
This question is meant to be done without computational aids.

Source: CMI enterance examination.

The answer is 4.

1 solution

A Former Brilliant Member
Jun 9, 2016

We note that $\sin x + \cos x = \sqrt2 \cos \left(x-\frac{\pi}{4} \right) \in [1, \sqrt2] \quad \forall x \in [0,1]$

Hence, as $x \in [0,1]$ , $10 x^{\sqrt2} \leq f(x) \leq 10 x$ With equality holding if and only if $x = 0 \: or \: 1$ .

As equality holds only for finitely many points, the inequalities become on strict integrating on all sides.

Hence, $4 < 10(\sqrt2 - 1) < \int_0^1 f(x) \, dx < 5 \\\implies \left \lfloor \int_0^1 f(x) \, dx\right \rfloor = \boxed{4}$

There's a more simpler method to do this. You can approximate the integral. You'll get something lesser than 5.

Aditya Kumar - 5 years ago

In the subjective exam, you don't have much time (I'm kinda slow in computations), plus if the method is non trivial, I'll have to provide a proof of it. This was what striked me first in the exam, so I went with it.

A Former Brilliant Member - 5 years ago

Ooh I didn't know that was a subjective exam.

Aditya Kumar - 5 years ago

@Aditya Kumar – There's no way to find the closed form of it right?

A Former Brilliant Member - 5 years ago

@A Former Brilliant Member – By real analysis I don't think so we can get a closed form. But by complex analysis there might be one. @Mark Hennings can help us.

Aditya Kumar - 5 years ago

@Aditya Kumar – Waiting for closed form.

A Former Brilliant Member - 5 years ago

@Aditya Kumar – I'm doubtful that a closed form can be found; I certainly don't know one. My solution to this problem was Deeparaj's...

Mark Hennings - 5 years ago

You can use am-gm inequality also right?

Abhi Kumbale - 4 years, 8 months ago

I'm not sure how you intend to achieve that. Could you post a solution that uses it?

A Former Brilliant Member - 4 years, 8 months ago

I can't post a solution write now as I do not have a laptop and I do not know latex. I'll try to post it as soon as possible. I think these problems might interest you,

https://brilliant.org/problems/check-this-out-3/?ref_id=1271985

https://brilliant.org/problems/convergency-and-divergency/?ref_id=1198750

https://brilliant.org/problems/think-this-is-easy/?ref_id=1260048

If u liked them I think you are welcome to try my set,

https://brilliant.org/profile/abhi-pwu19k/sets/my-creations-check-them-out/?ref_id=1198669

Thank you!!

Abhi Kumbale - 4 years, 8 months ago

Perfect solution!! Upvoted!!

rajdeep brahma - 3 years, 2 months ago

No computational aids required!

Source: CMI enterance examination.

The answer is 4.

1 solution

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