Determine whether Rolle's Theorem can be applied to f on the...

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 $f^{\prime}(c)=0 .$ If Rolle's Theorem cannot be applied, explain why not. $$f(x)=\left(x^{2}-2 x\right) e^{x},[0,2]$$

Key Concepts

Solving for the Zero of the Derivative

After computing the derivative of the function, the next key step is setting the derivative equal to zero and solving the resultant equation. This process involves algebraic manipulation to isolate the variable, revealing the value(s) of c that satisfy f?(c) = 0 in the open interval. The identification of such points confirms the behavior dictated by Rolle's Theorem.

Derivative Calculation Using the Product Rule

When a function is given as the product of two functions, the derivative is computed using the product rule. This rule states that the derivative of a product f(x)g(x) is f'(x)g(x) + f(x)g'(x). This method simplifies the differentiation of complex expressions and is essential for finding critical points where the derivative equals zero.

Continuity and Differentiability

For Rolle's Theorem to be applicable, the function must be continuous on the closed interval and differentiable on the open interval. Continuity ensures there are no sudden jumps or breaks in the graph, while differentiability guarantees the existence of a well-defined tangent (and thus derivative) at every point in the interval. These conditions are crucial in applying the theorem to identify points where the instantaneous rate of change (the derivative) is zero.

Rolle's Theorem

Rolle's Theorem is a fundamental result in calculus that guarantees, under certain conditions, the existence of at least one point c in the open interval (a, b) where the derivative of a function is zero, provided that the function is continuous on the closed interval [a, b], differentiable on (a, b), and its values at the endpoints are equal. This theorem is a special case of the Mean Value Theorem and is used to confirm the existence of stationary points in functions.

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