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Functions: 2012x Denotes Fractional Part Denotes Greatest Integer Funct

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

Question
If f(x)=cos⁡ ⁣(2010{x3}(2011[x2]+2012x))f(x) = \cos\!\left(2010\{x^3\}(2011^{[x^2]}+2012x)\right), x∈Rx \in \mathbb{R}, where {⋅}\{\cdot\} denotes fractional part and [⋅][\cdot] denotes greatest integer function, then the value of fmax⁡f_{\max}.
Solution
Answer: 1
Step 1: Identify when the cosine attains its maximum The maximum of cos⁡(⋅)\cos(\cdot) is 1, achieved when the argument equals 0. Step 2: Find xx for which the argument is 0 For x∈Zx \in \mathbb{Z}: x3x^3 is an integer, so {x3}=0\{x^3\} = 0, making the entire argument 00.
f(x)=cos⁡(0)=1f(x) = \cos(0) = 1
 Since cos⁡(⋅)≤1\cos(\cdot) \leq 1 for all inputs, fmax⁡=1f_{\max} = 1. Answer: 1
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