Problem 12 - Essential Calculus

Question

Determine whether Rolle's Theorem can be applied to \(f\) on the closed interval \([a, b] .\) If Rolle's Theorem can be applied, find all values of \(c\) in the open interval \((a, b)\) such that \(f^{\prime}(c)=0\). $$ f(x)=x^{2}-5 x+4,[1,4] $$

Step-by-Step Solution

Answer

Rolle's Theorem can be applied to the function. The value of \(c\) in the open interval (1,4) which satisfies \(f'(c) = 0\) is \(c = 2.5\).

Step 1: Verify Continuity

This is a polynomial function, which are continuous everywhere. So the function is continuous in the interval [1,4].

Step 2: Verify Differentiability

As a polynomial function, this function is differentiable everywhere, including the open interval (1,4).

Step 3: Verify Function Values at Endpoints

Calculate f(1) and f(4):\n f(1) = \(1^2 -5*1 +4 = 0\)\n f(4) = \(4^2 -5*4 +4 = 0\)\n Since they are equal, the function satisfies all the conditions of Rolle's Theorem.

Step 4: Find c

Differentiate the function: \n f'(x) = \(2*x -5\)\n Set f'(c) equal to zero and solve for c:\n 2*c -5 = 0 ==> c = 5/2. \n Hence, \(c = 2.5\) in the open interval (1, 4) satisfies f'(c) = 0.

Key Concepts

Polynomial Function

Polynomial functions are some of the most common types of algebraic expressions that students encounter. They are made up of terms that are the product of constants and variables raised to whole number powers. For instance, the function given in the problem, \(f(x) = x^2 - 5x + 4\), is a quadratic polynomial, because it has terms up to the second power of \(x\).
The structure of polynomial functions makes them very predictable and easy to work with:

• They have smooth, continuous graphs with no breaks or holes.
• The degree of the polynomial determines the number of turns the graph can have.
• Solutions to polynomial equations correspond to the x-intercepts of the graph, also known as roots.

Understanding polynomial functions and their properties is key to applying theorems like Rolle's Theorem effectively.

Continuity

Continuity is a fundamental property of functions, particularly when examining the application of Rolle's Theorem. A function is continuous on an interval if there are no breaks, jumps, or holes in its graph within that interval. In simpler terms, you can draw the graph of the function on that interval without lifting your pencil.
For a polynomial function like \(f(x) = x^2 - 5x + 4\), continuity is guaranteed everywhere, as polynomials are continuous by nature:

• They maintain a consistent graph with no interruptions.
• This continuous nature makes them predictable and reliable for analysis.

Rolle's Theorem requires that the function be continuous on the closed interval \([a, b]\), which is easily satisfied by any polynomial, including the one given in this exercise.

Differentiability

Differentiability pertains to the function's capability of having a derivative at any point in its domain. In simple words, if you can calculate the slope or the rate of change of the function at any point, then the function is differentiable at that point. Unlike continuity, differentiability is a bit stricter as it also requires the graph not to have any sharp turns or cusps.

For a polynomial function like \(f(x) = x^2 - 5x + 4\), differentiability is not a concern since polynomial functions are differentiable everywhere:

• Their graphs are smooth curves.
• You can compute derivatives for all points within the domain.

This property is important when using Rolle's Theorem, as it necessitates differentiability on the open interval \((a, b)\), which is met by the polynomial given.

Critical Point

A critical point of a function on a given interval is where the derivative equals zero or is undefined. For polynomial functions, critical points occur when the derivative equals zero, which can signal where the function's graph has a peak, valley, or flat area.
In this exercise, we determine the critical point by setting the derivative equal to zero. Given \(f'(x) = 2x - 5\), setting \(f'(c) = 0\) results in \(c = \frac{5}{2}\). This value of \(c\) is where the function's slope is zero, thereby fulfilling the condition for Rolle's Theorem.
The critical point \(c = 2.5\) falls within the open interval \((1, 4)\), indicating a horizontal tangent to the graph, which suggests either a local maximum, minimum, or a point of inflection without necessarily being one in particular.