If fx- 3 3x 3x2-2a2 3x 3x2-2a2 3x3-4a2x 3x2-2a2 3x3-6a2x 3x4-8a2x2-2a4- for some real constant a- then

F(x)=| nbsp;3 amp; 3x amp; 3x^2+2a^2 nbsp; nbsp; nbsp; nbsp; nbsp;3x amp; 3x^2+2a^2 amp; 3x^3+4a^2x nbsp; nbsp; nbsp; nbsp; ...

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Definition of Functions

If fx=| 3 3...

Question

If $f (x) = ∣ ∣ ∣ ∣ ∣ \begin{matrix} 3 & 3 x & 3 x^{2} + 2 a^{2} 3 x & 3 x^{2} + 2 a^{2} & 3 x^{3} + 4 a^{2} x 3 x^{2} + 2 a^{2} & 3 x^{3} + 6 a^{2} x & 3 x^{4} + 8 a^{2} x^{2} + 2 a^{4} \end{matrix} ∣ ∣ ∣ ∣ ∣$ for some real constant $a,$ then

f^{'} (x) = 0

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y = f (x)

is a straight line parallel to

x -

axis

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2 \int 0 f (x) d x = 32 a^{4}

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f (x) = - 16 a^{6}

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Solution

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f (x) = ∣ ∣ ∣ ∣ ∣ \begin{matrix} 3 & 3 x & 3 x^{2} + 2 a^{2} 3 x & 3 x^{2} + 2 a^{2} & 3 x^{3} + 4 a^{2} x 3 x^{2} + 2 a^{2} & 3 x^{3} + 6 a^{2} x & 3 x^{4} + 8 a^{2} x^{2} + 2 a^{4} \end{matrix} ∣ ∣ ∣ ∣ ∣

a,

f (x) = ∣ ∣ ∣ ∣ ∣ \begin{matrix} 3 & 3 x & 3 x^{2} + 2 a^{2} 3 x & 3 x^{2} + 2 a^{2} & 3 x^{3} + 6 a^{2} x 3 x^{2} + 2 a^{2} & 3 x^{3} + 6 a^{2} x & 3 x^{4} + 12 a^{2} x^{2} + 2 a^{4} \end{matrix} ∣ ∣ ∣ ∣ ∣

∣ ∣ ∣ ∣ ∣ \begin{matrix} 3 & 3 x & 3 x^{2} + 2 a^{2} 3 x & 3 x^{2} + 2 a^{2} & 3 x^{3} + 6 a^{2} x 3 x^{2} + 2 a^{2} & 3 x^{3} + 6 a^{2} x & 3 x^{4} + 12 a^{2} x^{2} + 2 a^{4} \end{matrix} ∣ ∣ ∣ ∣ ∣

A (x) = ∣ ∣ ∣ ∣ ∣ \begin{matrix} 3 & 3 x & 3 x^{2} + 2 a^{2} 3 x & 3 x^{2} + 2 a^{2} & 3 x^{3} + 6 a^{2} x 3 x^{2} + 2 a^{2} & 3 x^{3} + 6 a^{2} x & 3 x^{4} + 12 a^{2} x^{2} + 2 a^{4} \end{matrix} ∣ ∣ ∣ ∣ ∣,

a \in R

f (x) = \frac{x^{2} - 1}{x^{2} + 1}

f (x)

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