If the function , where a > 0, attains its local maximum and local minimum values at p and q, respectively, such that , then f(3) is equal to : [2025]

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Solution

Q.
If the function $f (x) = 2 x^{3} - 9 a x^{2} + 12 a^{2} x + 1$ , where a > 0, attains its local maximum and local minimum values at p and q, respectively, such that $p^{2} = q$ , then f(3) is equal to : [2025]

1 55

2 10

3 37

4 23

Ans.
(3)

We have, $f (x) = 2 x^{3} - 9 a x^{2} + 12 a^{2} x + 1$

$\Rightarrow f' (x) = 6 x^{2} - 18 a x + 12 a^{2}$

$\Rightarrow f (x) = 6 (x^{2} - 3 a x + 2 a^{2}) = 6 (x - a) (x - 2 a)$

$\Rightarrow f' (x) = 0 \Rightarrow 6 (x - a) (x - 2 a) = 0 \Rightarrow x = a, 2 a$

Now, $f'' (x) = 6 (2 x - 3 a)$

$f'' (a) = 6 (2 a - 3 a) = - 6 a < 0$

$∴$ x = a is point of maxima { $∵$ a > 0}

$f'' (2 a) = 6 (4 a - 3 a) = 6 a > 0$

$∴$ x = 2a is a point of minima

So, p = a and q = 2a.

Given, $p^{2} = q \Rightarrow a^{2} = 2 a \Rightarrow a = 2$

Now, $f (x) = 2 x^{3} - 9 a x^{2} + 12 a^{2} x + 1$

f(3) = 54 – 162 + 144 + 1 = 37.

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