Astudent claims that the function f(x)=x4+kx2+1 is an even function. which statement best describes the student's claim?

a) the student's claim is only true for even values of k.

b) the student's claim is only true for odd values of k.

c) the student's claim is true for all values of k.

d) the student's claim is not true for any value of k.

A

Step-by-step explanation:

Suppose x is odd. Then x^4 is also odd. So x^4 + 1 is even. Then for the whole function to be even, kx^2 should also be even. x^2 is obviously odd as x is odd. So to make kx^2 even, k should be even.

Now suppose x is even. Then x^4 + 1 is odd. So to make the function even, kx^2 should also be odd. But x is even, so kx^2 automatically becomes even, and then the whole function becomes odd. So the student's statement doesn't hold when x is even, no matter what the value of k is.

So the statement is true when x is odd and k is even.

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