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 \(f^{\prime}(c)=0\). \(f(x)=6\) on [-1,1] .

Short answer

Rolle's Theorem applies; any point \(c\) in \((-1,1)\) satisfies \(f'(c) = 0\).

Step-by-step solution

Verify Conditions for Rolle's Theorem

For Rolle's Theorem to be applicable, the function must be continuous on the closed interval \([-1, 1]\), differentiable on the open interval \((-1, 1)\), and \(f(-1) = f(1)\). Since \(f(x) = 6\) and 6 is a constant, it is continuous and differentiable everywhere, and \(f(-1) = f(1) = 6\). Thus, all conditions are satisfied for Rolle's Theorem.

Find the Derivative of the Function

Since \(f(x) = 6\) is a constant function, its derivative is \(f'(x) = 0\) everywhere.

Determine the Value of c

According to Rolle's Theorem, there exists at least one \(c\) in the interval \((-1, 1)\) such that \(f'(c) = 0\). In this particular case, because \(f'(x) = 0\) everywhere in \((-1, 1)\), every point in the interval satisfies \(f'(c) = 0\). Therefore, any \(c\) in \((-1, 1)\) satisfies the condition.

Key concepts

Continuous Function

A continuous function is a fundamental concept in calculus. It means that the function does not break, jump, or have any gaps in a given interval. Imagine you were drawing a graph of a function without lifting your pen from the paper; that's what continuity represents. For a function to be continuous on a closed interval \([a, b]\), it must be continuous at every point in between and also include the end points.
In the given exercise, the function is \({f(x) = 6}\), which is a constant function. Constant functions are always continuous over any interval because they do not change; they remain constant. Therefore, \({f(x) = 6}\) is continuous on the interval \([-1, 1]\). This fulfills the first condition necessary for applying Rolle's Theorem.

Differentiable Function

Differentiability is a measure of how a function can change. A function is differentiable over an interval if it has a derivative at each point in that interval, which means it has a well-defined slope (or tangent) everywhere. All differentiable functions are continuous, but not all continuous functions are differentiable.
In simpler terms, differentiability implies smoothness. There should be no sharp corners or cusps in the graph of the function. For the function \({f(x) = 6}\), it is differentiable everywhere since a constant has the same value at every point; hence, it has no variation or sharpness to its form. This confirms the second condition of Rolle's Theorem that \({f(x)}\) must be differentiable on \((-1, 1)\).

Constant Function

A constant function is quite simple but pivotal in understanding many calculus concepts. It is a type of function where the output value remains the same for any input value. Mathematically, a constant function can be expressed as \({f(x) = c}\), where \(c\) is a constant real number.

• It is always linear and parallel to the x-axis.
• Its graph is a straight horizontal line.

For the exercise, our constant function \({f(x) = 6}\) means no matter what \(x\) you input, the output is always six. This characteristic makes it straightforward to verify the conditions for Rolle's Theorem: \({f(c_1) = f(c_2)}\) for any \(c_1\) and \(c_2\) in \([-1, 1]\), including where \(f(c) = f(-1) = f(1) = 6\). This fulfills the need for the function values at the interval's endpoints to be equal.

Derivative

In calculus, the derivative of a function represents the rate at which a function is changing at any given point and is often described as the function's slope. Finding a derivative essentially gives us a new function that tells us how steep the original function is at each point.

• A derivative can be constant (if the original function is a straight line).
• It can vary (if the function has curves).

In our case, with \({f(x) = 6}\), the derivative is \({f'(x) = 0}\). This is because the value of the function does not change regardless of the x-value. The graph does not rise or fall; it stays perfectly flat. According to Rolle's Theorem, since \({f'(c) = 0}\) everywhere, particularly in the open interval \((-1, 1)\), any \(c\) within this interval will satisfy the theorem's conclusion.