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Functions: 2012x Denotes Fractional Part Denotes Greatest Integer Funct

JEE Maths question with a full step-by-step solution.

Question

f(x) = \cos\!\left(2010\{x^3\}(2011^{[x^2]}+2012x)\right)

x \in \mathbb{R}

, where

\{\cdot\}

denotes fractional part and

[\cdot]

denotes greatest integer function, then the value of

f_{\max}

Solution

Answer: 1

Step 1: Identify when the cosine attains its maximum The maximum of

\cos(\cdot)

is 1, achieved when the argument equals 0. Step 2: Find

x

for which the argument is 0 For

x \in \mathbb{Z}

x^3

is an integer, so

\{x^3\} = 0

, making the entire argument

0

f(x) = \cos(0) = 1

Since

\cos(\cdot) \leq 1

for all inputs,

f_{\max} = 1

. Answer: 1

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