Let a > 0. If the function attains its local maximum and minimum values at the points and respectively such that , then is equal to : [2025]

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Solution

Q.
Let a > 0. If the function $f (x) = 6 x^{3} - 45 a x^{2} + 108 a^{2} x + 1$ attains its local maximum and minimum values at the points $x_{1}$ and $x_{2}$ respectively such that $x_{1} x_{2} = 54$ , then $a + x_{1} + x_{2}$ is equal to : [2025]

1 18

2 24

3 13

4 15

Ans.
(1)

We have, $f (x) = 6 x^{3} - 45 a x^{2} + 108 a^{2} x + 1$ .

Since local max. and min. values occur when $f' (x) = 0$

$f' (x) = 18 x^{2} - 90 a x + 108 a^{2} = 0 \Rightarrow x = 2 a and 3 a$

i.e., $x_{1} = 2 a, x_{2} = 3 a$

Also, we have $x_{1} x_{2} = 54 \Rightarrow 6 a^{2} = 54 \Rightarrow a = 3$

$∴ a + x_{1} + x_{2} = 3 + 6 + 9 = 18$ .

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