Problem 33 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Let \(f(x)=1+3 x^{2}+3^{2} x^{4}+\ldots+3^{30} \cdot x^{60} .\) Then \(f(x)\) has (A) atleast one maximum (B) exactly one maximum (C) atleast one minimum (D) exactly one minimum

Step-by-Step Solution

Answer

(D) Exactly one minimum.

Step 1: Recognize the Pattern in the Series

The function \( f(x) = 1 + 3x^2 + 3^2x^4 + \ldots + 3^{30}x^{60} \) is a finite geometric series. Notice that it's in the form of \( \sum_{n=0}^{30} (3x^2)^n \).

Step 2: Recognize the Geometric Series Formula

Recall that the sum of a geometric series \( \sum_{n=0}^{N} a r^n \) is given by \( \frac{a(1-r^{N+1})}{1-r} \) for \( r eq 1 \). Here, \( a = 1 \) and \( r = 3x^2 \).

Step 3: Apply the Geometric Series Formula

Applying the geometric series formula, we get the sum \[ f(x) = \frac{1 - (3x^2)^{31}}{1 - 3x^2} \].

Step 4: Consider the Properties of f(x)

To determine the extrema of \( f(x) \), investigate where its derivative \( f'(x) \) is zero and its behavior as \( x \to \pm \infty \).

Step 5: Analyze the Behavior of f(x) as x Approaches Infinity

As \( x \to \infty \), \( 3x^2 \to \infty \), and hence the series behaves like \( \frac{1}{-3x^2} \), which approaches zero. Similarly, as \( x \to -\infty \), the behaviour is mirrored.

Step 6: Find the Critical Points by Derivation

Taking the derivative of \( f(x) \), set \( f'(x) = 0 \) to identify the critical points. This requires finding points where the numerator's derivative \( -(3x^2)^{31} \) becomes significant.

Step 7: Confirm the Nature of the Critical Point(s)

Given the nature of a finite geometric progression and polynomial structure, examine behavior near roots and endpoints to confirm whether maxima or minima exist.

Key Concepts

Understanding Extrema

Extrema, which consist of maximum and minimum values, are critical points of a function. These points are essential for understanding where a function achieves its highest or lowest values.
To determine extrema, you typically analyze the derivative of the function. A critical point is found where the derivative is zero or undefined.
For the given function, derivative analysis provides insights into the behavior of the function:

• Setting the derivative of the function to zero helps identify critical points.
• Examining the second derivative or combining it with other tests (like the First Derivative Test) can confirm the nature (maximum/minimum) of these critical points.

In the context of a geometric series, understanding extrema helps to determine how the series will behave, especially near its boundaries.

Exploring Finite Series

A finite series is a sum of numbers that terminates after a specific number of terms. This concept is crucial in calculus and algebra. Finite series differ from infinite series, which go on indefinitely.
In this exercise, the function is a finite geometric series composed of several terms:

• Starts with the term 1 and ends with the term of the form \((3^{30}x^{60})\).
• The formula used to calculate a finite geometric series is: \( \frac{a(1-r^{N+1})}{1-r} \), where \(a\) is the first term and \(r\) is the common ratio.

These characteristics make finite series valuable for precise calculations, such as when identifying sums up to a specified point, providing clear bounds for mathematical problems.

Insight into Geometric Progressions

A geometric progression is a sequence where each term is derived by multiplying the previous term by a constant called the common ratio. Understanding this structure helps in predicting the behavior of more complex functions.
In the case of the exercise, a geometric progression can be seen in the series:

• The first term is 1, and subsequent terms multiply by \(3x^2\), with the series not exceeding 31 terms.
• Offers a straightforward pattern making it easier to apply mathematical operations such as summation using the geometric series formula.

Recognizing geometric progressions is key for effectively dealing with algebraic expressions, allowing one to address problems regarding series convergence and behavior effectively.