Let f [a, b] → R and g [a, b] → R be continuous functions that are differentiable on (a, b), and let H [a, b] → R be given by
H(x) := (g(b)−g(a))*(f(x)−f(a)) − (f(b)−f(a))*(g(x)−g(a))
(a) Briefly explain why H satisfies all the hypotheses of Rolle’s theorem on [a, b].
b) Suppose that g′(x) not equal to 0 for any x ∈ (a,b). Briefly explain why this implies g(b) not equal to g(a). [Hint: MVT for g.]
c) till assuming g′(x) not equal to 0 for any x ∈ (a, b), apply Rolle’s theorem to H on [a,b] to draw a conclusion about the ratio: (f(b)-f(a)) / (g(b)-g(a))
lease help me with the following (thank you in advance)?
2017-01-30 8:42 am
回答 (2)
2017-01-30 10:17 am
a) H(a) = (g(b)−g(a))*(f(a)−f(a)) − (f(b)−f(a))*(g(a)−g(a)) = 0
H(b) = (g(b)−g(a))*(f(b)−f(a)) − (f(b)−f(a))*(g(b)−g(a)) = 0
H(x) is continuous and differentiable since H(x) = (g(b)−g(a))*(f(x)−f(a)) − (f(b)−f(a))*(g(x)−g(a))
because 2 terms on rhs are continuous and differentiable
So H(a) = H(b) = 0
and H(x) continuous on [a,b] and differentiable on (a,b)
so hypotheses of Rolle’s theorem on [a, b] are fulfilled
b) We have (g(b) - g(a))/(b-a) = g'(c) for some c on [a,b]
So if g'(c) = e ( e not equal to 0 on [a,b]
then g(b) - g(a) = e(b-a) and <=> 0
so g(b) <=> g(a)
H(b) = (g(b)−g(a))*(f(b)−f(a)) − (f(b)−f(a))*(g(b)−g(a)) = 0
H(x) is continuous and differentiable since H(x) = (g(b)−g(a))*(f(x)−f(a)) − (f(b)−f(a))*(g(x)−g(a))
because 2 terms on rhs are continuous and differentiable
So H(a) = H(b) = 0
and H(x) continuous on [a,b] and differentiable on (a,b)
so hypotheses of Rolle’s theorem on [a, b] are fulfilled
b) We have (g(b) - g(a))/(b-a) = g'(c) for some c on [a,b]
So if g'(c) = e ( e not equal to 0 on [a,b]
then g(b) - g(a) = e(b-a) and <=> 0
so g(b) <=> g(a)
2017-01-30 8:46 am
a) List the hypotheses of Rolle's theorem. "Let f be continuous on a closed interval [a, b] and differentiable on the open interval (a, b)".
Is H continuous on a closed interval [a,b]? Show why.
Is it differentiable on the open interval (a,b)? Show why.
Done.
b) Hint: MVT for g.
What does the MVT for g say? How does it relate g'(x) to g(b) - g(a)? Write it down. Look at the form "if p, then q". The equivalent statement is "if not q, then not p".
Is H continuous on a closed interval [a,b]? Show why.
Is it differentiable on the open interval (a,b)? Show why.
Done.
b) Hint: MVT for g.
What does the MVT for g say? How does it relate g'(x) to g(b) - g(a)? Write it down. Look at the form "if p, then q". The equivalent statement is "if not q, then not p".
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