Question

1. For each, circle either True or False. please add an explanation to each one so i can understand the reasoning  

(a) (1 point) A continuous function is always integrable. T/F

(b) (1 point) A differentiable function is always continuous. T/F

(c) (1 point) An integrable function is always differentiable. T/F

(d) (1 point) If c is a critical value of function f then f(c) is a relative maximum. T/F

(e) (1 point) If c is a relative minimum of function f then c is a critical value of f. T/F

(f) (1 point) If f is not differentiable at a then f(a) does not exist. T/F

(g) (1 point) f00(c) = 0 implies c is an infection point. T/F

(h) (1 point) A relative maximum is always an absolute maximum. T/F

Answer 1

A continuous function is always integrable. (TRUE)

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A differentiable function is always continuous. (TRUE)

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An integrable function is always differentiable (False)

Because we know that a function with a finite number of point of discontinuity is Riemann Integrable. But this discontinuous function is not differentiable because we know the discontinuous function is not differentiable.

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If c is a critical value of function f then f(c) is a relative maximum. (False)

Because of relative minimum depends upon the second-order derivative of that function.

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If c is a relative minimum of function f then c is a critical value of f. (True)

Because of the first condition for a point to be a relative minimum is to be a critical point.

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If f is not differentiable at a then f(a) does not exist. (False)

Counter Example: let, f(x)=|x| for all real number x.

Then the graph of f has a sharp point at (0,0)

We know that the existence of such a sharp point is the cause of not differentiability. i.e. f is not differentiable at x=0 but f(0)=0

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f00(c) = 0 implies c is an infection point.

for this statement ''f00(c)'' is not understandable.

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A relative maximum is always an absolute maximum. (False)