Corollary 5.3.3. If g: AR is differentiable on an interval A and satisfies g'(x) = 0 for all x = A, then g(x) = k for some constant k = R. Proof. Take x, y = A and assume x
Corollary 5.3.3. If g: AR is differentiable on an interval A and satisfies g'(x) = 0 for all x = A, then g(x) = k for some constant k = R. Proof. Take x, y = A and assume x
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter4: Polynomial And Rational Functions
Section4.5: Rational Functions
Problem 51E
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Please prove Corollary 5.3.4
![Corollary 5.3.3. If g: AR is differentiable on an interval A and satisfies
g'(x) = 0 for all x = A, then g(x) = k for some constant k = R.
Proof. Take x, y = A and assume x <y. Applying the Mean Value Theorem to
g on the interval [x, y], we see that
g'(c) =
g(y) − g(x)
y-x
for some c A. Now, g'(c) = 0, so we conclude that g(y) = g(x). Set k equal
to this common value. Because x and y are arbitrary, it follows that g(x) = k
for all x € A.
☐
Corollary 5.3.4. If ƒ and g are differentiable functions on an interval A and
satisfy f'(x) = g'(x) for all x E A, then f(x) = g(x) + k for some constant
k Є R.
Proof. Let h(x) = f(x) = g(x) and apply Corollary 5.3.3 to the differentiable
function h.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc6389447-1237-4af0-b5c6-eb1260425b55%2F8bfc6954-986b-4741-99db-75e61a5716da%2Fyng2lfb_processed.png&w=3840&q=75)
Transcribed Image Text:Corollary 5.3.3. If g: AR is differentiable on an interval A and satisfies
g'(x) = 0 for all x = A, then g(x) = k for some constant k = R.
Proof. Take x, y = A and assume x <y. Applying the Mean Value Theorem to
g on the interval [x, y], we see that
g'(c) =
g(y) − g(x)
y-x
for some c A. Now, g'(c) = 0, so we conclude that g(y) = g(x). Set k equal
to this common value. Because x and y are arbitrary, it follows that g(x) = k
for all x € A.
☐
Corollary 5.3.4. If ƒ and g are differentiable functions on an interval A and
satisfy f'(x) = g'(x) for all x E A, then f(x) = g(x) + k for some constant
k Є R.
Proof. Let h(x) = f(x) = g(x) and apply Corollary 5.3.3 to the differentiable
function h.
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