# Calculus Theorems

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2016-11-02 16:00

## Section

Question | Answer |
---|---|

Intermediate Value Theorem | If f is continuous on [a,b] and L is any number between f(a) and f(b), then there is a number c that is between a and b so that f(c)=L. |

Root Location Theorem | If f is a continuous function on [a,b] and if f(a) and f(b) have opposite signs, then f(c)=0 for at least one number c on the open interval (a,b). |

Extreme Value Theorem | If f is continuous on a closed and bounded interval [a,b] then f has both and absolute maximum and an absolute minimum. Absolute extrema of a continuous function on a closed interval exist. |

Critical Number Theorem | If a continuous function f has a relative extremum at x=c, then either f ' (c)=0 or f ' (c) does not exist. Absolute extrema can only occur at critical points or at the endpoints a,b. |

Rolle's Theorem | Let f be continuous on the closed interval [a,b] and differentiable on the open interval (a,b). If f(a) = f(b), then there is at least one number c in (a,b) such that f ' (c)=0. |

Mean Value Theorem | If f is continuous on the closed interval [a,b] and differentiable on the open interval (a,b), then there exists a least one number c in (a,b) such that [(f(b)-f(a))/(b-a)]=f ' (c) |

Zero-Derivative Theorem | If f is continuous on the closed interval [a,b] and differentiable on the open interval (a,b) with f ' (x) = 0 for all x in (a,b), then f is constant on [a,b] |

Constant Difference Theorem | Suppose f and g are continous on the closed interval [a,b] and differentiable on the open interval (a,b). If f ' (x) = g ' (x) for all x in (a,b), then there i a constant c so that f(x)=g(x)+c for all x on [a,b] |

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