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Given that \(f(x) = 2x^4 − x^2,\) and we need to find the value of \(f(2\sqrt{3})\)

To find the value of \(f(2\sqrt{3})\) we will compare what is inside the bracket in \(f(2\sqrt{3})\) and in f(x)
=> x = \(2\sqrt{3}\)

=> We can get value of \(f(2\sqrt{3})\) by putting x = \(2\sqrt{3}\) in f(x)
=> \(f(2\sqrt{3})\) = \(2*(2\sqrt{3})^4 - (2\sqrt{3})^2\)
= \(2 * 2^4 * 3^2 - 2^2*3\) = 2^5*9 - 4*3 = 32*9 - 12 = 288-12 = 276

So, Answer will be D
Hope it helps!

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\(f(x) = 2x^4 - x^2\)

\(f(2\sqrt{3}) = 2*(2\sqrt{3})^4 - (2\sqrt{3})^2\)

\(f(2\sqrt{3}) = 2*((2\sqrt{3})^2)^2 - (2\sqrt{3})^2\) (so that we can re-use the value of \((2\sqrt{3})^2\) and save ourselves calculations.)

\(f(2\sqrt{3}) = 2*((2\sqrt{3})^2)^2 - 12\)

\(f(2\sqrt{3}) = 2*(12^2) - 12\)

\(f(2\sqrt{3}) = 2*144 - 12\)

\(f(2\sqrt{3}) = 288 - 12\)

Now, at this point we stop, since 288 is one of the choices, and so the answer has to be less than 288. But clearly it cannot be 200 either, so we settle for Choice D.

If we calculate,

\(f(2\sqrt{3}) = 288 - 12\)

\(f(2\sqrt{3}) = 276\)

Hence, the answer is Choice D.