Value of function

Question

If !!f(x) = cos[pi^2]x + cos[-pi^2]x!!

Here [!!" "!!] represents !!GIF!!.

Then,

!!1)" "f(pi/2) = 1/2!!

!!2)" "f(pi) = 1!!

!!3)" "f(pi/4) = 1!!

!!4)" "f(-pi) = 0!!

Solution

If !!f(x) = cos[pi^2]x + cos[-pi^2]x!!

Here we need to use our concepts of greatest integer function.

!!If f(x) = cos[3.14^2]x + cos[-(3.14)^2]x!!

!!If f(x) = cos[9.86]x + cos[-9.86]x!!

We know that for positive values in !!GIF!! we need to ignore decimal value and take only integer value and for negative values in !!GIF!! we need to take one number lower than integer value.

Hence,

!!If f(x) = cos(9)x + cos(-10)x!!

!!If f(x) = cos9x + cos10x!!

Now we can check for the function.

!!(1)" "f(pi/2) = cos9(pi/2) + cos10(pi/2) = cos(pi/2) + cos5pi = 0 -1 = -1 !! So this does not satisfy.

!!(2)" "f(pi) = cos9(pi) + cos10(pi) = -1 + 1 = 0!!

This also does not satisfy.

!!(3)" "f(pi/4) = cos9(pi/4) + cos10(pi/4) = 1/sqrt2!!

!!(4)" "f(-pi) = cos9(-pi) + cos10(-pi) = cos9(pi) + cos10(pi) = -1 + 1 = 0!!

Hence, fourth option is correct.

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