(a) Suppose that f is differentiable on R and has two roots. Show that has at least one root.
(b) Suppose f is twice differentiable on R and has three roots. Show that has at least one real root.
(c) Can you generalize parts (a) and (b)?

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Nestor RutherfordLv2

21 May 2020

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5. (3 points) Which of the following statements best describes the relationship between continuity and differentiability for a function? Choose only one, and assume that the functions are defined for all real numbers. (A) A function must be both differentiable and continuous at r = a or neither; you cannot have one without the other. (B) If a function is differentiable at I = a then it must also be continuous at r = a. (C) If a function is continuous at r= a then it must also be differentiable at r=a. (D) There is at least one function that is continuous for some r = a where it is not differentiable, and at least one other function that is differentiable at some r = a where it is not continuous. (E) Two of the above statements mean the same thing, and are both the correct choice.

(a) Prove that the equation has at least one real root.

(b) Use your graphing device to find the root correct to three decimal places.