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

f^{'} (x) = 0

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y = f (x)

is a straight line parallel to x-axis

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

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none of these

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Solution

The correct options are
A $f^{'} (x) = 0$
B $y = f (x)$ is a straight line parallel to x-axis
$∣ ∣ ∣ ∣ ∣ \begin{matrix} 3 & 3 x & 3 x^{2} + 2 a^{2} 3 x & 3 x^{2} + 2 a^{2} & 3 x^{3} + 6 a^{2} x 3 x^{2} + 2 a^{2} & 3 x^{3} + 6 a^{2} x & 3 x^{4} + 12 a^{2} x^{2} + 2 a^{4} \end{matrix} ∣ ∣ ∣ ∣ ∣$
Applying, $R_{2} = R_{2} - x R_{1}$ ,
$= ∣ ∣ ∣ ∣ ∣ \begin{matrix} 3 & 3 x & 3 x^{2} + 2 a^{2} 0 & 2 a^{2} & 4 a^{2} x 3 x^{2} + 2 a^{2} & 3 x^{3} + 6 a^{2} x & 3 x^{4} + 12 a^{2} x^{2} + 2 a^{4} \end{matrix} ∣ ∣ ∣ ∣ ∣$
Applying, $C_{3} = C_{3} - 2 x C_{2}$ ,
$= ∣ ∣ ∣ ∣ ∣ \begin{matrix} 3 & 3 x & 2 a^{2} - 3 x^{2} 0 & 2 a^{2} & 0 3 x^{2} + 2 a^{2} & 3 x^{3} + 6 a^{2} x & 2 a^{4} - 3 x^{4} \end{matrix} ∣ ∣ ∣ ∣ ∣$
Expanding by $R_{2}$ ,
$= 2 a^{2} (6 a^{4} - 9 x^{4} - 4 a^{4} + 9 x^{4}) = 2 a^{2} (2 a^{4}) = 4 a^{6}$
So, $f^{'} (x) = 0$
$y = f (x) = 4 a^{6} =$ constant, so, it will be parallel to x-axis

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Similar questions

∣ ∣ ∣ ∣ ∣ \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} ∣ ∣ ∣ ∣ ∣

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} + 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,

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

\int f (x) d x = f (x)

\int (\frac{f (x)}{f^{'} (x)})

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