\displaystyle 1.\ \text{Rolle's Theorem: Let } f \text{ be a real valued function defined on the closed interval }[a,b]\ \text{such that}
\displaystyle \text{(i) it is continuous on }[a,b]\ \text{(ii) it is differentiable on }(a,b),\ \text{and (iii) } f(a)=f(b).
\displaystyle \text{Then, there exists a real number } c\in(a,b)\ \text{such that } f'(c)=0.
\displaystyle \text{Geometrical Interpretation: Let } f(x)\ \text{be a real valued function defined on }[a,b]\ \text{such} \\ \text{that the curve } y=f(x)\ \text{is a continuous curve between points } A(a,f(a))\ \text{and } B(b,f(b)) \\ \text{and the curve has a unique tangent at every point between } A \text{ and } B.\ \text{Also, the} \\ \text{ordinates at the end points of the interval }[a,b]\ \text{are equal. Then there exists at least} \\ \text{one point } (c,f(c)) \text{between } A \text{ and } B \text{ on the curve where tangent is parallel to } x\text{-axis.}
\displaystyle \text{Algebraic Interpretation: Between any two roots of a polynomial } f(x),\ \text{there is} \\ \text{always a root } \text{sof its derivative.}

\displaystyle 2.\ \text{Lagrange's Mean Value Theorem: Let } f(x)\ \text{be a function defined on }[a,b]\ \text{such that it} \\ \text{is continuous on }[a,b]\ \text{and differentiable on }(a,b).\ \text{Then, there exists } c\in(a,b)\ \text{such that}
\displaystyle f'(c)=\frac{f(b)-f(a)}{b-a}.
\displaystyle \text{Geometrical Interpretation: Let } f(x)\ \text{be a function defined on }[a,b]\ \text{such that the curve } \\ y=f(x)\ \text{is a continuous curve between points } A(a,f(a))\ \text{and } B(b,f(b))\ \text{and at every} \\ \text{point on the curve, except at the end-points, it is possible to draw a unique tangent.} \\ \text{Then there exists a point on the curve such that tangent at it is parallel to the chord} \\ \text{joining the end points of the curve.}

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