Question

If the function \( f(x) = 2x^3 - 9ax^2 + 12a^2x + 1 \), where \( a>0 \), attains its local maximum and minimum at \( p \) and \( q \), respectively, such that \( p^2 = q \), then \( f(3) \) is equal to:

• A. 23
• B. 55
• C. 10
• D. 37

Hint:

Critical points from \( f'(x) = 0 \) help determine max/min values. Use given condition on them.

Answer 1

The Correct Option is D

The objective is to determine the value of \( f(3) \) for the function \( f(x) = 2x^3 - 9ax^2 + 12a^2x + 1 \), where \( a>0 \). It is given that the function's local maximum and minimum occur at points \( p \) and \( q \) respectively, and these points satisfy the condition \( p^2 = q \).

Methodology:

The process for finding local extrema of a differentiable function involves:

1. First Derivative Test: Local extrema are found at critical points where the first derivative is zero or undefined. For polynomials, this means solving \( f'(x) = 0 \).

2. Second Derivative Test: This test classifies critical points. If \( c \) is a critical point where \( f'(c) = 0 \):

• If \( f''(c) < 0 \), then \( f(x) \) has a local maximum at \( x = c \).
• If \( f''(c) > 0 \), then \( f(x) \) has a local minimum at \( x = c \).

Solution Steps:

The function is \( f(x) = 2x^3 - 9ax^2 + 12a^2x + 1 \).

The first derivative is:

\[ f'(x) = 6x^2 - 18ax + 12a^2 \]

Setting \( f'(x) = 0 \) to find critical points:

\[ 6x^2 - 18ax + 12a^2 = 0 \]

Dividing by 6 yields:

\[ x^2 - 3ax + 2a^2 = 0 \]

Factoring the quadratic equation:

\[ (x - a)(x - 2a) = 0 \]

The critical points are \( x = a \) and \( x = 2a \). These are \( p \) and \( q \).

The second derivative is:

\[ f''(x) = 12x - 18a \]

Evaluating \( f''(x) \) at the critical points:

At \( x = a \):

\[ f''(a) = 12a - 18a = -6a \]

Since \( a > 0 \), \( f''(a) < 0 \), indicating a local maximum at \( x = a \). Thus, \( p = a \).

At \( x = 2a \):

\[ f''(2a) = 12(2a) - 18a = 24a - 18a = 6a \]

Since \( a > 0 \), \( f''(2a) > 0 \), indicating a local minimum at \( x = 2a \). Thus, \( q = 2a \).

Using the given condition \( p^2 = q \):

\[ (a)^2 = 2a \implies a^2 - 2a = 0 \]

Factoring gives \( a(a - 2) = 0 \). Since \( a > 0 \), we have \( a = 2 \).

Substituting \( a = 2 \) into \( f(x) \):

\[ f(x) = 2x^3 - 18x^2 + 48x + 1 \]

Final Calculation:

Calculating \( f(3) \):

\[ f(3) = 2(3)^3 - 18(3)^2 + 48(3) + 1 \] \[ f(3) = 2(27) - 18(9) + 144 + 1 \] \[ f(3) = 54 - 162 + 144 + 1 = 37 \]

The value of \( f(3) \) is 37.