Maths Q8 – JEE-MAIN 2025

Home Maths Continuity and DifferentiabilityQuestion #152

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 :

Clue 1 of 3: Critical Points

To find local maximum and minimum, first find the critical points by setting the first derivative of the function to zero.

🧠 Full Solution Path

● Video Walkthrough

Step 1: Find Critical Points✦ Active

First, find the derivative of the function $f (x)$ and set it to zero to find the critical points.

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

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

6 x^{2} - 18 a x + 12 a^{2} = 0

x^{2} - 3 a x + 2 a^{2} = 0

(x - a) (x - 2 a) = 0

The critical points are $x = a$ and $x = 2 a$ .

Step 2: Identify Maxima/Minima and Solve for 'a'○ Expand

Use the second derivative test to determine which critical point corresponds to the local maximum and which to the local minimum. Then, apply the given condition $p^{2} = q$ to find the value of $a$ .

f^{″} (x) = 12 x - 18 a

At $x = a$ : $f^{″} (a) = 12 a - 18 a = - 6 a$ . Since $a > 0$ , $f^{″} (a) < 0$ , so $x = a$ is a local maximum. Thus, $p = a$ .

At $x = 2 a$ : $f^{″} (2 a) = 12 (2 a) - 18 a = 24 a - 18 a = 6 a$ . Since $a > 0$ , $f^{″} (2 a) > 0$ , so $x = 2 a$ is a local minimum. Thus, $q = 2 a$ .

Given the condition $p^{2} = q$ :

a^{2} = 2 a

Since $a > 0$ , we can divide by $a$ , which gives $a = 2$ .

💡 Teacher's Secret Hint

Remember that $a > 0$ is a crucial condition for simplifying $a^{2} = 2 a$ to $a = 2$ (otherwise $a = 0$ would also be a solution).

Step 3: Calculate

f (3)

○ Expand

Substitute the value of $a = 2$ back into the original function $f (x)$ and then evaluate $f (3)$ .

f (x) = 2 x^{3} - 9 (2) x^{2} + 12 (2^{2}) x + 1

f (x) = 2 x^{3} - 18 x^{2} + 48 x + 1

f (3) = 2 (3)^{3} - 18 (3)^{2} + 48 (3) + 1

f (3) = 2 (27) - 18 (9) + 144 + 1

f (3) = 54 - 162 + 144 + 1

f (3) = 199 - 162

f (3) = 37

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Maths Question 8 – JEE-MAIN 2025

Clue 1 of 3: Critical Points

🧠 Full Solution Path