Problem 161 - A Complete Resource Book in Mathematics for JEE Main

Question

If the function \(f(x)=2 x^{3}-9 a x^{2}+12 a^{2} x+1\), where \(a>0\), attains its maximum and minimum at \(p\) and \(q\) respectively such that \(p^{2}=q\), then \(a\) equals \(\quad\) [2003] (A) 3 (B) I (C) 2 (D) \(\frac{1}{2}\)

Step-by-Step Solution

Answer

Insufficient steps provided.

Step 1: Find the derivative

To find the maximum and minimum of the function, start by calculating the first derivative. The function given is \(f(x) = 2x^3 - 9ax^2 + 12a^2x + 1\). The first derivative is obtained by differentiating each term: \(f'(x) = \frac{d}{dx}(2x^3) - 9a\frac{d}{dx}(x^2) + 12a^2\frac{d}{dx}(x)\). Applying the power rule \(f'(x) = 6x^2 - 18ax + 12a^2\).

Step 2: Find the first derivative

Compute \(f'(x)\) using the appropriate differentiation rules.

Step 3: Find critical points

Set \(f'(x) = 0\) and solve for \(x\). Also check where \(f'(x)\) is undefined.

Step 4: Apply the second derivative test

Compute \(f''(x)\) and evaluate it at each critical point to classify them as local maxima, minima, or neither.

Step 5: Determine the answer

Based on the analysis, identify the required extrema and state the final answer.

Key Concepts

Differentiation

Differentiation is a fundamental concept in calculus, which enables us to find the rate at which a function is changing at any given point. It involves determining the derivative of a function, which tells us how fast the function's output value is changing as its input value changes. Differentiation is especially useful for finding maxima and minima of functions, as these points are where the function changes direction - from increasing to decreasing, or vice versa.
To differentiate a function, you typically find the first derivative, denoted as \( f'(x) \). This involves differentiating each term of the function with respect to the variable, often using rules like the power rule. The process can be thought of as finding the slope of the tangent line at any point on the function's graph.

• The first derivative can help identify critical points, where the derivative is zero or undefined.
• At critical points, the function may have relative maxima, minima, or a saddle point.

Power Rule

The power rule is a powerful tool used in differentiation, especially when dealing with polynomial functions. It simplifies the process of finding derivatives of terms where the variable is raised to a power. The general form of the power rule states that if you have a term \( x^n \), its derivative is \( nx^{n-1} \).
Consider differentiating a function, such as \( f(x) = 2x^3 - 9ax^2 + 12a^2x + 1 \). Applying the power rule allows each term to be individually differentiated easily:

• For \( 2x^3 \), the derivative is \( 6x^2 \).
• For \( -9ax^2 \), the derivative is \( -18ax \).
• For \( 12a^2x \), the derivative is \( 12a^2 \).

Once you've applied the power rule to each term, you can determine the first derivative of the whole function, which is crucial for finding maxima and minima.

Polynomial Functions

Polynomial functions are expressions formed by terms containing variables raised to whole number powers, combined with coefficients. An example of a polynomial function is \( f(x) = 2x^3 - 9ax^2 + 12a^2x + 1 \), which is a cubic polynomial. These functions are defined for all real numbers and are known for their smooth, continuous curves.
Analyzing polynomial functions to find their maxima and minima involves understanding their structure and behavior. Usually, the degree of the polynomial indicates the maximum number of roots or solutions it can have. A cubic polynomial, for instance, can have up to three roots.

• The end behavior of polynomial functions depends on the leading term (the term with the highest power).
• Polynomial functions can model various real-world scenarios, making them extremely valuable in mathematical problem-solving.

By differentiating polynomial functions, you can locate the points where the slope is zero, which are possible locations for maxima and minima.