Q
jee_main_2025_02_april_morning
Maxima and Minima
If the function f(x) = 2x^3 - 9ax^2 + 12a^2x + 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:
Solution
### Related Formula
For local extrema of a differentiable function, set the first derivative to zero:
f'(x) = 0
### Core Logic
Differentiate f(x) to obtain its critical points p and q in terms of a, use the constraint p^2 = q to fix a, and evaluate f(3).
### Step 1: Differentiate and Find Critical Points
f'(x) = 6x^2 - 18ax + 12a^2
Set f'(x) = 0:
6(x^2 - 3ax + 2a^2) = 0 implies 6(x-a)(x-2a) = 0
The critical points are x = a and x = 2a. Since a > 0, checking the sign change of f'(x) shows that the local maximum occurs at the smaller root (p = a) and the local minimum at the larger root (q = 2a).
### Step 2: Apply Root Constraint
Given p^2 = q:
a^2 = 2a implies a(a-2) = 0
Since a > 0, we get a = 2.
### Step 3: Evaluate function at x = 3
Substitute a = 2 back into the definition of f(x):
f(x) = 2x^3 - 18x^2 + 48x + 1
Now compute f(3):
f(3) = 2(3)^3 - 18(3)^2 + 48(3) + 1 = 54 - 162 + 144 + 1 = 37
### Pattern Recognition
For cubic equations with two distinct real critical roots, the smaller root is always the local maximum if the leading coefficient is positive (2 > 0). This ensures p=a and q=2a$ immediately without relying on secondary derivative checks.
### Evaluation Rubric / Model Answer
null
### Chapter Mix
Class 12 Mathematics: Application of Derivatives
