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.