Question

A function \(f(x)\) and interval \([a, b]\) are given. Check if Rolle's Theorem can be applied to fon \([a, b] ;\) if so, find cin \([a, b]\) such that \(f^{\prime}(c)=0\). \(f(x)=x^{2}+x-2\) on [-3,2]

Short answer

Rolle's Theorem applies, and \( c = -\frac{1}{2} \).

Step-by-step solution

Verify continuity on [a, b]

The function \( f(x) = x^2 + x - 2 \) is a polynomial, which is continuous everywhere. Hence, \( f(x) \) is continuous on the interval \([-3, 2]\).

Verify differentiability on (a, b)

As a polynomial, \( f(x) \) is also differentiable everywhere, particularly on the open interval \((-3, 2)\). Thus, \( f(x) \) is differentiable on \((-3, 2)\).

Check if f(a) = f(b)

Calculate \( f(-3) \) and \( f(2) \):\[\begin{align*}f(-3) &= (-3)^2 + (-3) - 2 = 9 - 3 - 2 = 4, \f(2) &= 2^2 + 2 - 2 = 4 + 2 - 2 = 4.\end{align*}\]Since \( f(-3) = f(2) = 4 \), \( f(a) = f(b) \).

Apply Rolle's Theorem

Since all conditions for Rolle's Theorem are satisfied—\( f(x) \) is continuous on \([-3, 2]\), differentiable on \((-3, 2)\), and \( f(-3) = f(2) \)—there exists at least one \( c \) in \( (-3, 2) \) such that \( f'(c) = 0 \).

Find c in (a, b) such that f'(c) = 0

First, find \( f'(x) \): \[ f'(x) = 2x + 1. \]Set the derivative to zero to find \( c \):\[ 2x + 1 = 0 \]Solve for \( x \):\[ 2x = -1 \] \[ x = -\frac{1}{2}. \]Therefore, \( c = -\frac{1}{2} \), which is in the interval \((-3, 2)\).

Key concepts

Polynomials

Polynomials are fundamental mathematical expressions that consist of variables and coefficients, arranged as a sum of terms. Each term is composed of a coefficient multiplied by a variable raised to a non-negative integer power. For instance, in the polynomial function \( f(x) = x^2 + x - 2 \), the terms are \( x^2 \), \( x \), and \( -2 \).
These types of functions are known for their smooth continuous curves when graphed. A key feature of polynomials is their behavior across all real numbers, meaning they do not have breaks or holes. This property makes polynomials widely applicable in calculus.
They also hold great significance in Rolle's Theorem because polynomials are automatically continuous and differentiable, fitting well within the domain of this theorem's application.
Understanding polynomials and their structures helps in recognizing the types of functions suitable for the implementation of mathematical theorems like Rolle's.

Continuity

Continuity refers to a function's property where its graph is an unbroken curve. For a function to be continuous over an interval, it must have no gaps, jumps, or points of discontinuity in that range. In simpler terms, you can draw the graph of a continuous function without lifting your pen.
Polynomials, such as \( f(x) = x^2 + x - 2 \), are inherently continuous everywhere. This means that for any given stretch on the x-axis, these functions smoothly transition through values without causing disruptions.
When applying Rolle's Theorem to any function, demonstrating continuity on the closed interval \([a, b]\) is the first crucial step. This verification assures us that the function operates without interruption over the interval, fulfilling one of the theorem's necessary conditions.

Differentiability

A function is said to be differentiable at a certain point if it has a derivative there. Essentially, this means that the function has a defined slope or rate of change at that point. Differentiability implies smoothness, with no sharp angles or breaks in the graph near the point in question.
For Rolle's Theorem, the function must be differentiable on the open interval \((a, b)\). Given the polynomial \( f(x) = x^2 + x - 2 \), this function is differentiable across all real numbers due to its smooth curve nature and lack of vertical tangents or cusps.
Ensuring differentiability in the interval confirms that we can find a derivative function whose value we then further manipulate to find points where the slope or derivative is zero, necessary for identifying critical points in Rolle's Theorem.

Critical Points

Critical points are the values of \( x \) in the domain of a function where its derivative is zero or undefined. For differentiable functions, critical points are where the slope of the tangent is flat (the derivative equals zero). These points often represent local maxima or minima and are essential in calculus for finding where changes in increasing or decreasing behavior occur.
In the context of Rolle's Theorem, once continuity and differentiability are verified and \( f(a) = f(b) \) is established, the theorem assures at least one critical point in the open interval \((a, b)\).
For the function \( f(x) = x^2 + x - 2 \), after computing the derivative \( f'(x) = 2x + 1 \) and setting it to zero, we solve to find a critical point at \( x = -\frac{1}{2} \). This value lies within the given interval \((-3, 2)\), hence fulfilling Rolle's Theorem requirements and demonstrating how to locate these crucial points mathematically.