Question 2

Geometry Level 3

Find the total number of values of x x where the function f ( x ) = ∑ n = 1 999 cos ⁡ ( n x ) f(x)= \sum_{n=1}^{999}\cos(\sqrt{n}x) attains its maximum value.

This question is part of the set For the JEE-nius .
2 1 None of these 4

Samarpit Swain by Samarpit Swain , 23, India

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1 solution

The maximum output of cos(x) is equal to 1 for any real x. In the given function, we get each term in the series = 1 if we have x=0. For other x, it won't be 1 for all the terms, it'll be less than it . So we have only one value for maximum out. Note: Notice that √n*x can't be an integral multiple of 2π for all terms

6 Helpful 0 Interesting 0 Brilliant 0 Confused

Why can't x be equal to 2 π n \frac{2\pi}{\sqrt{n}} ??? Is there any thing wrong in that ?

Arnav Das - 4 years, 6 months ago

( 2 π ) / n (2 \pi)/\sqrt{n} (or 2\pi * \sqrt{n}) will be integral for the nth term, but not for most other terms. For example, there is no value of x ≠ 0 x\neq 0 such that x ∗ 3 x*\sqrt{3} and x ∗ 5 x*\sqrt{5} are both integral multiples of 2 ∗ π 2*\pi

Richard Desper - 4 years, 6 months ago

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