Calculus DifferentiabliityIf  is continuous on [4,7] which of the following statement must be true?   I      has a maximum value on [4,7] II      has a minimum value on [4,7] III  ...

Calculus Differentiabliity

If  is continuous on [4,7] which of the following statement must be true?

 

I      has a maximum value on [4,7]

II      has a minimum value on [4,7]

III    f(7)>f(4) .

IV Lim as x approaches 0 f(x)=f(6)

 

 

Asked on by hellome234

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embizze's profile pic

embizze | High School Teacher | (Level 1) Educator Emeritus

Posted on

Given that `f(x)` is continuous on [4,7]:

(1),(2) `f(x)` has a maximum and a minimum on [4,7] is true. The Extreme Value Theorem state that if f is a continuous function on a closed interval then it achieves both its maximum and minimum at least once in the interval.

(3) `f(7)>f(4) ` is false. We are not told that `f` is an increasing function.

(4) `lim_(x->0)f(x)=f(6)` is false. If you meant `lim_(x->6)f(x)=f(6)` then this is true -- a function is continuous at a point `c` if `lim_(x->c)f(x)` exists, `f(c)` exists, and `lim_(x->c)f(x)=f(c)` .

nathanshields's profile pic

nathanshields | High School Teacher | (Level 1) Associate Educator

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If a function is continuous on an interval, then a particular value is assigned to the function for each x value within the interval.  So both I and II are true.

Incidentally, this is not the case if you just say "f is continuous," which implies "f is continuous on its domain."  For example, f(x) = 1/x is continuous, but does not have a maximum or minimum value.

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