Q: State And Prove Rolle's Theorem With Diagram?
Rolle's Theorem states that if a function \( f(x) \) is continuous on a closed interval \([a, b]\), differentiable on the open interval \((a, b)\), and \( f(a) = f(b) \), then there exists at least one value \( c \) in \((a, b)\) such that \( f'(c) = 0 \).
To prove Rolle's Theorem, we will use the following steps:
1. Continuity: First, we need to show that the function \( f(x) \) is continuous on the closed interval \([a, b]\). Since this is given in the statement of the theorem, we don't need to prove it.
2. Differentiability: Next, we need to prove that the function \( f(x) \) is differentiable on the open interval \((a, b)\). This means that the derivative \( f'(x) \) exists for every value in \((a, b)\). We can use the differentiability of \( f(x) \) to prove this.
3. Equal function values: Since \( f(a) = f(b) \), it means that the function has the same value at the endpoints of the interval.
4. Application of Mean Value Theorem (MVT): By applying the MVT, we know that there exists at least one value \( c \) in \((a, b)\) where the derivative \( f'(c) \) is equal to the average rate of change of the function. The average rate of change is given by \( \frac{f(b) - f(a)}{b - a} = 0 \), since \( f(a) = f(b) \). Therefore, \( f'(c) = 0 \).
The diagram below illustrates the function \( f(x) \) on the interval \([a, b]\):
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a b
In the diagram, the function starts at a point on the left and ends at the same point on the right, with possible ups and downs in between. The point where \( f'(c) = 0 \) is represented by the point marked with an asterisk. This point is guaranteed to exist according to Rolle's Theorem.
Rolle's theorem is a fundamental theorem in calculus that relates to the behavior of a real-valued differentiable function on a closed interval. Rolle's theorem states that if a function is continuous on a closed interval [a, b], differentiable on the open interval (a, b), and the function values at both endpoints of the interval are equal (f(a) = f(b)), then there must exist at least one point c in the open interval (a, b) where the derivative of the function is equal to zero (f'(c) = 0).
To prove Rolle's theorem, we will follow these steps:
Step 1: Verify that the function satisfies the necessary conditions of Rolle's theorem.
Step 2: Show that the function is continuous on the closed interval [a, b].
Step 3: Show that the function is differentiable on the open interval (a, b).
Step 4: Show that the function values at both endpoints of the interval are equal (f(a) = f(b)).
Step 5: Conclude that there exists at least one point c in the open interval (a, b) where the derivative of the function is equal to zero (f'(c) = 0).
Now, let's visualize this with a diagram:
```
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x----------------------------------
a c b
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In this diagram, we have a function represented by the curve. The points a and b are the endpoints of the closed interval [a, b]. The point c represents the point in the open interval (a, b) where the derivative of the function is equal to zero.
By following the steps above and examining the diagram, we can understand and prove Rolle's theorem.
To state and prove Rolle's Theorem, we need to understand its conditions and requirements.
Rolle's Theorem states that if a function f(x) is continuous on a closed interval [a, b] and differentiable on the open interval (a, b), and if f(a) = f(b), then there exists a number c in the open interval (a, b) such that the derivative of f(x) at c, denoted as f'(c), is equal to zero.
Now let's prove this theorem step by step:
Step 1: Continuity
Firstly, we need to establish that the function f(x) is continuous on the interval [a, b]. To prove continuity, we can consider the following:
- f(x) is defined and exists for every value in the interval [a, b].
- The limit of f(x) as x approaches any value in the interval [a, b] exists and is finite.
- The function f(x) has no jumps, holes, or vertical asymptotes within the interval [a, b].
Step 2: Differentiability
Next, we need to show that the function f(x) is differentiable on the open interval (a, b). This means that the derivative of f(x), denoted as f'(x), exists and is finite for every value of x within the interval (a, b). We can use the definition of differentiability to establish this.
Step 3: Checking the Endpoints
Since the theorem states that f(a) = f(b), we need to check the values of f(x) at the endpoints a and b. If f(a) = f(b), then we have established the necessary condition for the existence of a point c. If f(a) ≠ f(b), then we cannot apply Rolle's Theorem.
Step 4: Finding the Zero-Derivative Point
Since the conditions of continuity and differentiability are satisfied, we know that the function f(x) has a derivative for every value of x within the open interval (a, b). By the Mean Value Theorem (a consequence of Rolle's Theorem), we know that there exists a point c in the open interval (a, b) such that the derivative of f(x) at c, denoted as f '(c), is equal to f(b) - f(a) divided by b - a (slope of the secant line). In this case, f(b) - f(a) equals 0 because f(a) = f(b) according to the theorem's condition. As a result, f '(c) = 0.
Diagram:
A diagram is not necessary to prove Rolle's Theorem since it is a result derived from the Mean Value Theorem using the concept of continuous and differentiable functions.