verify rolle's theorem for function
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Rolle’s theorem. If ff is continuous on [a,b][a,b]and differentiable on (a,b)(a,b), with f(a)=f(b)f(a)=f(b), then there exists c∈(a,b)c∈(a,b) such that f(c)=0f(c)=0.
By the way, such a point cc can be chosen to be either a point of absolute minimum or a point of absolute maximum for ff in (a,b)(a,b).
The given function satisfies the hypotheses: over (−3,3)(−3,3) we have f(x)=9−x2f(x)=9−x2 which is clearly differentiable. The function is continuous over [−3,3][−3,3] by continuity of the absolute value.
In this case there is just one point where the derivative vanishes, namely x=0x=0.
However this is not general: the function sinxsinxover [0,2π][0,2π] has the derivative that vanishes at two points. Over the interval [0,nπ][0,nπ] the derivative vanishes at nn points (nn positive integer).
The function σσ defined by
σ(x)=⎧⎩⎨0xsinπxif x=0if x≠0σ(x)={0if x=0xsinπxif x≠0
satisfies the hypotheses of Rolle’s theorem over [0,1][0,1]; there are infinitely many points in (0,1)(0,1)where the derivative vanishes.
By the way, such a point cc can be chosen to be either a point of absolute minimum or a point of absolute maximum for ff in (a,b)(a,b).
The given function satisfies the hypotheses: over (−3,3)(−3,3) we have f(x)=9−x2f(x)=9−x2 which is clearly differentiable. The function is continuous over [−3,3][−3,3] by continuity of the absolute value.
In this case there is just one point where the derivative vanishes, namely x=0x=0.
However this is not general: the function sinxsinxover [0,2π][0,2π] has the derivative that vanishes at two points. Over the interval [0,nπ][0,nπ] the derivative vanishes at nn points (nn positive integer).
The function σσ defined by
σ(x)=⎧⎩⎨0xsinπxif x=0if x≠0σ(x)={0if x=0xsinπxif x≠0
satisfies the hypotheses of Rolle’s theorem over [0,1][0,1]; there are infinitely many points in (0,1)(0,1)where the derivative vanishes.
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