Problem 4 - The Calculus with Analytic Geometry

Question

Verify that conditions (i), (ii), and (iii) of the hypothesis of Rolle's theorem are satisfied by the, given function on the indicated interval. Then find a suitable value for \(c\) that satisfies the conclusion of Rolle's theorem $$ f(x)=x^{3}-16 x ;[-4,0] $$

Step-by-Step Solution

Answer

All conditions of Rolle's theorem are satisfied, and the value of \( c \) is \( -\frac{4\dotsqrt3}{3} \).

Step 1: Verify Continuity

Check if the function \( f(x) = x^3 - 16x \) is continuous on the interval \([-4,0]\). Since \( f(x) = x^3 - 16x \) is a polynomial, it is continuous everywhere, including \([-4,0]\). Thus, condition (i) is satisfied.

Step 2: Verify Differentiability

Check if the function is differentiable on the interval \((-4,0)\). Since \( f(x) = x^3 - 16x \) is a polynomial, it is differentiable everywhere, including \((-4,0)\). Thus, condition (ii) is satisfied.

Step 3: Check Endpoints

Evaluate the function at the endpoints of the interval to verify \( f(-4) = f(0) \). Calculate \( f(-4) = (-4)^3 - 16(-4) = -64 + 64 = 0 \). Calculate \( f(0) = 0^3 - 16(0) = 0 \). Both evaluations give 0, so \( f(-4) = f(0) \). Thus, condition (iii) is satisfied.

Step 4: Apply Rolle's Theorem

Since \( f(x) \) satisfies all conditions of Rolle's theorem, there exists at least one \( c \in (-4,0) \) where \( f'(c) = 0 \). First, compute the derivative \( f'(x) \). \( f'(x) = \frac{d}{dx} (x^3 - 16x) = 3x^2 - 16 \). Set the derivative equal to zero and solve for \( x \): \( 3x^2 - 16 = 0 \ \Rightarrow x^2 = \frac{16}{3} \ \Rightarrow x = \pm \frac{4}{\sqrt{3}} = \pm \frac{4\sqrt{3}}{3} \). Since we need \( c \in (-4,0) \), the suitable value is \( c = -\frac{4\dotsqrt,3}{3} \).

Key Concepts

Continuity

In mathematics, continuity is an essential property of functions. A function is continuous if, intuitively, you can draw its graph without lifting your pen from the paper. Mathematically, a function is continuous at a point if the limit of the function as it approaches that point is equal to the function's value at that point. This means:Whenever the input values move closer to a certain value, the output (or function values) also get closer to a specific value, confirming the function doesn't 'jump' or break at that point.
For our polynomial function \( f(x) = x^3 - 16x \), it's crucial to check continuity over the interval \([-4,0]\).
Luckily, all polynomials are smooth and continuous everywhere on their domain. This makes checking continuity for polynomials straightforward.
Since \( f(x) \) is a polynomial, it's continuous on the entire real line, including the interval \([-4,0]\). Thus, we have verified condition (i) of Rolle's theorem.

Differentiability

Differentiability is another vital property in calculus. A function is differentiable at a point if it has a defined derivative at that point. In simple terms, this means the function's graph has a tangent line at that point and doesn't have any sharp angles or cusps there.
More formally, a function is differentiable at a point if its derivative exists at that point. If a function is differentiable at every point in an interval, it's called differentiable on that interval.
For our given function \( f(x) = x^3 - 16x \), we need to check if it's differentiable on the open interval \((-4,0)\).
Just like continuity, all polynomial functions are differentiable everywhere. So, our function \( f(x) \) is differentiable over the entire interval \((-4,0)\), verifying condition (ii) of Rolle's theorem.

Polynomials

Polynomials are some of the most fundamental and simple functions in mathematics. They are expressions involving variables and coefficients, with non-negative integer powers of the variables.
A general polynomial has the form \( P(x)= a_n x^n + a_{(n-1)} x^{(n-1)} + ... + a_1 x + a_0 \), where \( a_n, a_{(n-1)}, ..., a_1, a_0 \) are constants, and \( n \) is a non-negative integer.
Key properties of polynomials include:

• Continuity: Polynomials are continuous for all real values of \( x \).


• Differentiability: Polynomials are differentiable everywhere, meaning the derivative exists for all real values of \( x \).


• Simplicity: Working with polynomials involves straightforward algebraic operations like addition, subtraction, multiplication, and differentiation.

Our example function \( f(x) = x^3 - 16x \) is a polynomial of degree 3. It smoothly transitions without any breaks, jumps, or sharp turns, making it very user-friendly for calculus operations like checking Rolle's theorem.