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Question: If \[f\left( x \right)=\cos \left[ {{\pi }^{2}} \right]x+\cos \left[ -{{\pi }^{2}} \right]x\] where ...

If f(x)=cos⁡[π2]x+cos⁡[−π2]xf\left( x \right)=\cos \left[ {{\pi }^{2}} \right]x+\cos \left[ -{{\pi }^{2}} \right]x where [.] stands for the greatest integer function, then which of the following is wrong.
A. f(π4)=1f\left( \dfrac{\pi }{4} \right)=1
B. f(π2)=−1f\left( \dfrac{\pi }{2} \right)=-1
C. f(π)=0f\left( \pi \right)=0
D. f(2π)=2f\left( 2\pi \right)=2

Explanation

Solution

At first, we find out the value of π2{{\pi }^{2}} which is 9.8699.869 . Now, we take the box function of 9.8699.869 and then −9.869-9.869 . The values come out as 9,−109,-10 respectively. We now simplify the function to f(x)=cos⁡9x+cos⁡10xf\left( x \right)=\cos 9x+\cos 10x . Putting the options one by one, we check which one is false.

Complete step by step solution:
The greatest integer function, also sometimes known as the box function is a certain type of function which returns an integer value as the output. To be precise, the box function returns an integer which is less than or equal to the input number. For example, let us consider the function y=[x]y=\left[ x \right] . If we take the value of x as 6.9526.952 , then the value of y will be 66 . Similarly, if we take the value of x to be −6.952-6.952 , the value of y comes out to be −7-7 .
Moving to the problem, the value of π2{{\pi }^{2}} is 9.8699.869 . This means that,
[π2]=9 [−π2]=−10 \begin{aligned} & \left[ {{\pi }^{2}} \right]=9 \\\ & \left[ -{{\pi }^{2}} \right]=-10 \\\ \end{aligned}
The function thus becomes,

& \Rightarrow f\left( x \right)=\cos 9x+\cos \left( -10 \right)x \\\ & \Rightarrow f\left( x \right)=\cos 9x+\cos 10x \\\ \end{aligned}$$ Now, we one by one put the options and check whether the values are correct or not. $f\left( \dfrac{\pi }{4} \right)=\cos 9\left( \dfrac{\pi }{4} \right)+\cos 10\left( \dfrac{\pi }{4} \right)=\dfrac{1}{\sqrt{2}}$ which is not equal to $1$ . **So, the correct answer is “Option A”.** **Note:** Greatest integer function and Lowest integer functions are quite confusing. So, we should learn their definitions by heart. Also, remembering their graphs also helps out to visualise the functions. We should also check the remaining options and verify whether all of them are correct or not.