The function in which the conditions of Rolle's theorem are...

The function in which the conditions of Rolle's theorem are satisfied, is
(A) $f(x)=2 x^{3}+x^{2}-4 x-2,[-\operatorname{root} 2$, root 2$]$
(B) tanx in $[0, \pi]$
(C) $f(x)=\left\{\begin{array}{l}x^{2}+1 ; 0 \leq x \leq 1 \\ 3-x ; 1 \leq x \leq 2\end{array}\right.$
(D) None of these

Key Concepts

Equal Endpoint Values

Equal endpoint values refer to the condition f(a) = f(b) in Rolle's Theorem. This requirement ensures that the function starts and ends at the same value over the interval, creating the necessary scenario where, due to continuity and differentiability, there must exist at least one point in between where the instantaneous rate of change (the derivative) is zero.

Differentiability

Differentiability means that a function has a defined derivative at every point in the interval under consideration (usually the open interval (a, b) for Rolle's Theorem). This property implies that the function's graph has a well-defined tangent at each point in the interval, which is critical for ensuring that a point exists where the derivative is exactly zero.

Continuity

Continuity refers to a function having no breaks, jumps, or holes over the interval in which it is defined, particularly on the closed interval [a, b]. For Rolle's Theorem to be applicable, the function must be continuous at every point within the interval, ensuring that the function can be drawn without lifting the pen, which is necessary to apply the intermediate value property.

Rolle's Theorem

Rolle's Theorem is a fundamental result in calculus which states that if a function is continuous on a closed interval [a, b], differentiable on the open interval (a, b), and has equal values at the endpoints (i.e., f(a) = f(b)), then there exists at least one point c in (a, b) at which the derivative of the function is zero. This theorem guarantees a stationary point under these conditions and serves as a precursor to the Mean Value Theorem.

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