Explain why or why not Determine whether the following statements are true and g

Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. If f'(X) > 0 and f"(x) 0 and f"(c) = 0, then f has a local maximum at c. Two functions that differ by an additive constant both increase and decrease on the same intervals. If f and g increase on an interval, then the product fg also increases on that interval. There exists a function f that is continuous on (-infinity, infinity) with exactly three critical points, all of which correspond to local maxima.

Explanation / Answer

a. True

Since f'(x) >0 , the function f(x) must be increasing. Since f''(x) <0, this means that f'(x) is decreasing, and thus the rate of increase of f(x) is decreasing.

b. false

f'(c) >0 means that at the point x=c , the function is increasing. f''(c) =0 means that at point x=c , the rate of increase of function has stopped. Thus function may be increasing but its rate of increase is constant.Since the function is still increasing , the point x=c cannot be a local maxima.

c. True
Since the function are only differed by a constant value the structure of the curve will be same . The functions will look like transaltions of each other. Since the function is only translated , the increase and decrease interval will remain the same.

d. False

If both function are increasing the product of function , we will presume that will also be increasing . Let's test this out. Consider the product of function f and g on a given interval to be h.

h(x) = f(x)*g(x)

The rate of increase of h(x) will be given by,

h'(x) = f'(x)*g(x) + f(x)*g'(x)

Now from here we can see that if we have a function whose value is negative on the interval but is increasing may result in a product which is decreasing. Consider on the interval [-1,0] two functions, y=x and y=x+0.5. Now both these function are increasing in the interval [-1,0] , but their product y=x(x+0.5) is increasing in (-0.25,0] but decreasing in [-1,-0.25). Thus the statement given must be false.

e. False

If there exists a local maxima then the points on the both side of the maxima will have to be smaller than the maxima. Thus before we encounter a maxima again on any side, we must face a minima first. Thus all three points cannot be maxima.

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