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Question

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

Accepted Answer

$$\begin{aligned}
&f(x)=6 x^3-45 a x^2+108 a^2 x+1\
&\text { For maxima or minima } f^{\prime}(x)=0
\end{aligned}$$

$$\begin{aligned}
& x_1 x_2=\frac{108 a^2}{18}=54 \
& \Rightarrow a^2=9 \Rightarrow a=3 \
& \text { Now, } a+x_1+x_2=3+\frac{90}{6}=3+15=18
\end{aligned}$$

Solutions · 1