Problem 115 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

If the equation \(a_{n} x^{n}+a_{n-1} x^{n-1}+\ldots \ldots \ldots . .+a_{1} x=0\) \(a_{1} \neq 0, n \geq 2\), has a positive root \(x=\alpha\), then the equation \(n a_{n} x^{n-1}+(n-1) a_{n-1} x^{n-2}+\ldots \ldots .+a_{1}=0\) has a positive root, which is (a) greater than \(\alpha\) (b) smaller than \(\alpha\) (c) greater than or equal to \(\alpha\) (d) equal to \(\alpha\)

Step-by-Step Solution

Answer

(b) smaller than \(\alpha\)

Step 1: Understand the given polynomial

The given polynomial is \(a_{n} x^{n}+a_{n-1} x^{n-1} + \ldots + a_{1} x = 0\). It is specified that \(a_{1} eq 0\) and the polynomial has a positive root \(x = \alpha\).

Step 2: Differentiate the polynomial

To find the derivative of the given polynomial, differentiate each term with respect to \(x\):\[\frac{d}{dx}(a_{n} x^{n}) = n a_{n} x^{n-1},\frac{d}{dx}(a_{n-1} x^{n-1}) = (n-1) a_{n-1} x^{n-2},\ldots\frac{d}{dx}(a_{1} x) = a_{1}\] Therefore, the derivative is \[n a_{n} x^{n-1} + (n-1) a_{n-1} x^{n-2} + \ldots + a_{1} = 0\].

Step 3: Apply Rolle's Theorem

Rolle's Theorem states if a function is continuous on \([a, b]\) and differentiable on \((a, b)\), with \(f(a) = f(b)\), then there is at least one \(c\) in \((a, b)\) such that \(f'(c) = 0\). Since the original polynomial has a positive root \(\alpha\), and the function is continuous and differentiable, it implies there must be a positive root to the derivative which lies between two positive roots of the original polynomial.

Step 4: Determine nature of the roots

Since the derivative has a positive root different from \(\alpha\) by Rolle's Theorem, this new root will be less than \(\alpha\), as it lies in the interval where the original function crosses from positive to negative or vice versa before reaching \(\alpha\).

Step 5: Conclusion

Based on the application of Rolle's Theorem and the properties of differentiable functions, the new polynomial formed by differentiating has a root smaller than the positive root \(\alpha\) of the original polynomial.

Key Concepts

Rolle's Theorem

Rolle's Theorem is a fundamental concept in calculus, often used to find roots of polynomial derivatives. Imagine it this way: if you have a smooth curve that starts and ends at the same point on a graph, there must be at least one spot along the curve where the slope, or tangent, is completely flat. This is the crux of Rolle's Theorem. It provides a bridge for understanding the behavior of polynomial functions.

• The theorem applies to functions that are continuous and differentiable.
• The key condition is that the start and end points of the interval must have the same function value.

When applied to polynomials, Rolle's Theorem assures us that if a polynomial has multiple roots, the derivative of the polynomial will have at least one root in the same interval. This is because the slopes (derivative values) must cross zero at least once somewhere between these roots. Thus, in solving polynomial equations, knowing Rolle's Theorem can hint at where new roots might appear in differentiated forms.

Positive Root Analysis

Positive root analysis involves examining where a polynomial equation can have solutions that are greater than zero. To unwrap the mystery of positive roots, start by recognizing that if a polynomial has a positive root, the function must curve up through the x-axis somewhere in the positive part of the graph.

• Since the polynomial's behavior around roots can indicate the presence of additional roots, evaluating the derivative helps us understand the nature around these points.
• Through analysis, the presence of a positive root can suggest sign changes in the graph and potential additional roots in close proximity.

When analyzing the derivative of the polynomial given in the problem, by observing the sign changes, we can determine that there must be other roots and also infer their magnitude relative to the original. Since Rolle's Theorem tells us of new roots smaller than the existing positive root, positive root analysis can predict behavior patterns and relationships between the roots.

Polynomial Roots

Understanding polynomial roots is key to delving into polynomial functions. Roots are where the polynomial equals zero; visually, they're points where the graph of the polynomial touches or crosses the x-axis.

• In our case, polynomial roots are especially interesting when they are positive since they craft specific characteristics of the function curve.
• Polynomial roots are influenced by the coefficients and the degree of the equation.

When differentiating, the resulting polynomial, through the derivative, may offer us new insights into other possible roots. These roots may shift, decrease, or even introduce altogether new roots as seen in polynomial differentiations like this example. The differentiated polynomial presents roots often tied directly to the changes in the slopes of the original function.