Let $\mathrm{a} \gt 0$. If the function $\mathrm{f}(\mathrm{x})=6 \mathrm{x}^3-45 \mathrm{a} \mathrm{x}^2+108 \mathrm{a}^2 \mathrm{x}+1$ attains its local maximum and minimum values at the points $x_1$ and $x_2$ respectively such that $x_1 x_2=54$, then $\mathrm{a}+\mathrm{x}_1+\mathrm{x}_2$ is…

Question

Let $\mathrm{a} \gt 0$. If the function $\mathrm{f}(\mathrm{x})=6 \mathrm{x}^3-45 \mathrm{a} \mathrm{x}^2+108 \mathrm{a}^2 \mathrm{x}+1$ attains its local maximum and minimum values at the points $x_1$ and $x_2$ respectively such that $x_1 x_2=54$, then $\mathrm{a}+\mathrm{x}_1+\mathrm{x}_2$ is equal to :-

Aicharya · Accepted Answer

$\begin{aligned} & f^{\prime}(x)=18 x^2-90 a x+108 a^2=0 \ & x=2 a \& x=3 a \ & x_1=2 a \quad x_2=3 a \ & x_1 x_2=54 \ & 6 a^2=54 \ & a=3 \ & a+x_1+x_2 \ & 3+2 	imes 3+3 	imes 3=18 \ & 	ext { option }(2)\end{aligned}$

Let $\mathrm{a} \gt 0$. If the function $\mathrm{f}(\mathrm{x})=6 \mathrm{x}^3-45 \mathrm{a}…

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