Question

please help!!

If the graphs of two differentiable functions f(x) and g(x) start at the same point in the plane and the functions have the same rate of change at every point, do the graphs have to be identical? Give reasons for your answer A corollary of the Mean Value Theorem states that if f7x): g7x) at each point x in an open interval (a,b), then there exists a constant C such that f(x)= g(x)-C for all Xe(a,b). That is, f-g is a constant function on (a,b) Identify the hypothesis of the corollary. Choose the correct answer below O A. At a point x in the interval (a,b), f(x) -g(x) O B. At each point x in an interval (a,b), f'(x)-g'(x) O C. The function f-g is a constant function on (a,b) O D. There exists a constant C such that f(x) g(x) C for all xE(a,b) Which given fact satisfies the corollary's hypothesis? Choose the correct answer below O A. The graphs of f and g have the same rate of change at every point. O B. The graphs of f(x) and g(x) start at the same point in the plane Since the hypothesis of the corollary is satisfied, which of the following conclusions may be drawn? Choose the correct answer below OA. At each point x in an interval-0o,0o), f'(x)-g'(x) O B. Both functions f and g are constant functions on (-0o,00) O C. The function f-g is a constant function on (- oo,00) O D. At a point x in the interval (-o0,00), f(x) g(x) Use the fact that at a starting point in the plane f(x) g(x), and the conclusion of the corollary to determine the value of f g Since the hypothesis of the corollary is satisfied, which of the following conclusions may be drawn? Choose the correct answer below A. At each point x in an interval (- oo, 0o), f'(x)-g'(x) O B. Both functions f and g are constant functions on(-00,00) ° C. The function f-g is a constant function on (-00,0) D. At a point x in the interval (-0o, 0o), f(x)-g(x) Use the fact that at a starting point in the plane fx) g(x), and the conclusion of the corollary to determine the value of f-g Do the graphs of f and g have to be identical? Choose the correct answer below. O A. No, because f'(x)-g'(x) B. No, because even though the difference between f and g is always zero, they do not start at the same point. ° C. Yes, because they start at the same point, and the difference between f and g always zero O D. Yes, because although they do not start at the same point, the difference between f and g is always zero. Since the hypothesis of the corollary is satisfied, which of the following conclusions may be drawn? Choose the correct answer below A. At each point x in an interval (- oo, 0o), f'(x)-g'(x) O B. Both functions f and g are constant functions on(-00,00) ° C. The function f-g is a constant function on (-00,0) D. At a point x in the interval (-0o, 0o), f(x)-g(x) Use the fact that at a starting point in the plane fx) g(x), and the conclusion of the corollary to determine the value of f-g Do the graphs of f and g have to be identical? Choose the correct answer below. O A. No, because f'(x)-g'(x) B. No, because even though the difference between f and g is always zero, they do not start at the same point. ° C. Yes, because they start at the same point, and the difference between f and g always zero O D. Yes, because although they do not start at the same point, the difference between f and g is always zero.

Answer 1

We are given that the function f and g have the same rate of change at all the points and they have one common value at one point.So let this point be x=a then we have f(a)=g(a) and f'(x) =g'(x) for all x.

Hence as given by the corollary of Mean Value Theorem we have f(x)=g(x)+c for some c, constant.

But we have f(a)=g(a). Hence putting thus value we get f(a)=g(a)+c which gives us f(a)=f(a)+c and hence we get c=0.

Thus the two functions are equal at all points.

Now we try to find the correct options for the given questions using the above and the given statements in the questions.

we have :

A corollary of the Mean Value Theorem states that if f'(x)=g'(x) at each point x in an open interval (a,b) ,then there is a constant C such that f(x)=g(x)+C for all x $\in$ (a,b).

Qu 1. Identify the hypothesis of the corollary. Choose the correct answer below.

Ans-Since the hypothesis means the information we assume to derive at a result,

hence

option (B) At each point x in an interval (a,b) f'(x)=g'(x) is the correct option.

Qu 2. Which given fact satisfies the corollary's hypothesis? Choose the correct answer below.

Ans-Since the rate of change of a function is the derivative of the function and we have derivatives are equal,

hence option (A)The graphs of f and g have the same rate of change at every point. is the correct option.

Qu 3. Since the hypothesis of the corollary is satisfied, which of the following conclusions may be drawn? Choose the correct answer below.

Ans-Clearly the conclusion we have is f(x)=g(x)+C which is f-g is constant.

Hence option (C)The function f-g is a constant function on
oc.. oc
is the correct option.

Qu 4. Use the fact that at a starting point in the plane f(x)=g(x), and the conclusions of the corollary to determine the value of f-g.

f-g=0,

Do the graphs of f and g have to be identical? Choose the correct option below.

Ans-We have proved that if the constant C is zero and hence the graphs are equal.

Hence option (C) Yes, because they start at the same point, and the difference between f and g is always zero. is the correct option.