Problem 40 - Single Variable Calculus

Question

Complete the proof of Rolle's Theorem by considering the case in which \(f(x)

Step-by-Step Solution

Answer

Define a new function \(g(x) = f(x) - d\), which is continuous and differentiable. Since \(f(x)

Step 1: Define the function g(x)

Define a new function g(x) = f(x) - d. Since f(x) is continuous and differentiable, g(x) is also continuous and differentiable.

Step 2: Determine g(a) and g(b)

Since f(x)

Step 3: Apply the Intermediate Value Theorem

Since g(x) is continuous on the closed interval [a, b] and there are non-negative values g(a) or g(b), the Intermediate Value Theorem implies that there exists a number x1 in the open interval (a, b) where g(x1) = 0, which means, f(x1) = d.

Step 4: Apply Fermat's Theorem

Fermat's Theorem states that if a function has a local extremum at a point within its domain, then the derivative at that point must be zero. Since g(x1) = 0 and g(x) is differentiable, it has a local extremum at x1. Hence, g'(x1) = 0.

Step 5: Determine g'(x) and f'(c)

Now, g'(x) = f'(x) because g(x) = f(x) - d, so g'(x1) = f'(x1). From Step 4, we know that g'(x1) = 0, which means f'(x1) = 0.

Step 6: Conclude the proof

We have found a number x1 (or c) in the open interval (a, b) such that f'(x1) = 0. This completes the proof of Rolle's Theorem by considering the case where f(x) < d for some number x in (a, b).

Key Concepts

Intermediate Value Theorem

The Intermediate Value Theorem is a fundamental concept in calculus. It is often used to show that a continuous function takes on every value between its endpoints over a closed interval. Imagine you're hiking up a hill and then down another; the theorem says you'll be at every height between the base and the top.

More formally, if you have a function, say \( f(x) \), that is continuous on a closed interval \([a, b]\), and \( f(a) \leq N \leq f(b) \), or \( f(a) \geq N \geq f(b)\), then there’s at least one number \( c \) in the interval \((a, b)\) where \( f(c) = N \).

In Rolle's Theorem, the Intermediate Value Theorem ensures us there is a point where the function reaches a particular value. In our exercise, use it to find a point \( x_1 \) where the function value is exactly \( d \). This particular application identifies points where specific conditions within the theorem are satisfied.

Fermat's Theorem

Fermat's Theorem is all about points where functions change their direction. If you think of the peak or valley of a function graph, those crest or trough points are what we consider the critical points, where the slope is flat.

Theorem states: if a function \( f(x) \) has a local maximum or minimum at some point \( c \), and if the derivative \( f'(x) \) exists, then the derivative at that point is zero, i.e., \( f'(c) = 0 \). This means the tangent to the graph at \( c \) is horizontal.

In the context of this exercise, once \( g(x_1) = 0 \) is found using the Intermediate Value Theorem, if \( x_1 \) yields a local extremum for \( g(x) \), then by Fermat's Theorem, its derivative is zero: \( g'(x_1) = 0 \). This is the pivotal step in concluding Rolle’s Theorem.

Continuity

Imagine a smooth, unbroken line that you can draw without lifting a pencil. That's essentially what continuity in mathematics looks like. A function is continuous if there are no sudden jumps, breaks, or holes in its graph at any point within its domain.

Formally, a function \( f(x) \) is continuous at a point \( x = c \) if:

• The function is defined at \( c \).
• The limit as \( x \) approaches \( c \) exists.
• The limit as \( x \) approaches \( c \) equals \( f(c) \).

For Rolle’s Theorem to work, the function has to be continuous over the closed interval \([a, b]\). This guarantees that it doesn't 'skip' values, which is crucial for applying the Intermediate Value Theorem.

Differentiability

Differentiability refers to the existence of a derivative for a function at a particular point. If a function is differentiable at a point \( c \), it means the function has a well-defined tangent at that point, or in simple terms, it has a slope there.

A differentiable function is also continuous, but the converse is not always true. Think of differentiability as continuity plus smoothness—no sharp turns or cusps.

For the case of Rolle’s Theorem, our function must be differentiable across an open interval \((a, b)\). Differentiability assures us that we can apply Fermat's Theorem, implying that the derivative is zero at any extremum found within this interval. This is key to proving that the zero derivative point \( x_1 \) exists.

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Problem 40

Other exercises in this chapter

Problem 40

$$ \text { Let } g(x)=\frac{\cos x}{\sqrt{x}} . \text { Find } \lim _{x \rightarrow \infty} g(x) \text { . } $$

View solution

Problem 40

Find the relative extrema of the function. $$ f(x)=3 \tan ^{-1} x-2 x $$

View solution

Problem 40

Find the relative extrema, if any, of the function. Use the Second Derivative Test, if applicable. $$ f(x)=2 x^{4}-8 x+4 $$

View solution

Problem 40

In Exercises \(25-40\), find the critical number \((s)\), if any, of the function. $$ g(t)=t^{2} \ln t $$

View solution

Key Concepts

Intermediate Value Theorem

The Intermediate Value Theorem is a fundamental concept in calculus. It is often used to show that a continuous function takes on every value between its endpoints over a closed interval. Imagine you're hiking up a hill and then down another; the theorem says you'll be at every height between the base and the top.

More formally, if you have a function, say \( f(x) \), that is continuous on a closed interval \([a, b]\), and \( f(a) \leq N \leq f(b) \), or \( f(a) \geq N \geq f(b)\), then there’s at least one number \( c \) in the interval \((a, b)\) where \( f(c) = N \).

In Rolle's Theorem, the Intermediate Value Theorem ensures us there is a point where the function reaches a particular value. In our exercise, use it to find a point \( x_1 \) where the function value is exactly \( d \). This particular application identifies points where specific conditions within the theorem are satisfied.

Fermat's Theorem

Fermat's Theorem is all about points where functions change their direction. If you think of the peak or valley of a function graph, those crest or trough points are what we consider the critical points, where the slope is flat.

Theorem states: if a function \( f(x) \) has a local maximum or minimum at some point \( c \), and if the derivative \( f'(x) \) exists, then the derivative at that point is zero, i.e., \( f'(c) = 0 \). This means the tangent to the graph at \( c \) is horizontal.

In the context of this exercise, once \( g(x_1) = 0 \) is found using the Intermediate Value Theorem, if \( x_1 \) yields a local extremum for \( g(x) \), then by Fermat's Theorem, its derivative is zero: \( g'(x_1) = 0 \). This is the pivotal step in concluding Rolle’s Theorem.

Continuity

Imagine a smooth, unbroken line that you can draw without lifting a pencil. That's essentially what continuity in mathematics looks like. A function is continuous if there are no sudden jumps, breaks, or holes in its graph at any point within its domain.

Formally, a function \( f(x) \) is continuous at a point \( x = c \) if:

• The function is defined at \( c \).
• The limit as \( x \) approaches \( c \) exists.
• The limit as \( x \) approaches \( c \) equals \( f(c) \).

For Rolle’s Theorem to work, the function has to be continuous over the closed interval \([a, b]\). This guarantees that it doesn't 'skip' values, which is crucial for applying the Intermediate Value Theorem.

Differentiability

Differentiability refers to the existence of a derivative for a function at a particular point. If a function is differentiable at a point \( c \), it means the function has a well-defined tangent at that point, or in simple terms, it has a slope there.

A differentiable function is also continuous, but the converse is not always true. Think of differentiability as continuity plus smoothness—no sharp turns or cusps.

For the case of Rolle’s Theorem, our function must be differentiable across an open interval \((a, b)\). Differentiability assures us that we can apply Fermat's Theorem, implying that the derivative is zero at any extremum found within this interval. This is key to proving that the zero derivative point \( x_1 \) exists.