2) Let f be continuous on [a, b], differentiable on (a, b) and positive (i.e., > 0) for all f'(c) (a, b) such that 1 = a-c x = (a, b). Prove that there exists c f(c) + b²c. (Hint: consider the function F(x) = (x − a) (x − b) ƒ (x) and use MVT for F (x) to show the existence of such c = (a, b).)
2) Let f be continuous on [a, b], differentiable on (a, b) and positive (i.e., > 0) for all f'(c) (a, b) such that 1 = a-c x = (a, b). Prove that there exists c f(c) + b²c. (Hint: consider the function F(x) = (x − a) (x − b) ƒ (x) and use MVT for F (x) to show the existence of such c = (a, b).)
Chapter3: Functions
Section3.3: Rates Of Change And Behavior Of Graphs
Problem 2SE: If a functionfis increasing on (a,b) and decreasing on (b,c) , then what can be said about the local...
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![2) Let f be continuous on [a, b], differentiable on (a, b) and positive (i.e., > 0) for all
1
x = (a, b). Prove that there exists c = (a, b) such that-
f'(c) 1
f(c) a-c
=
+
(Hint:
b-c
(x − a) (x − b) ƒ (x) and use MVT for F(x) to show
consider the function F(x)
the existence of such c = (a, b).)
3) Use L'Hopital's Rule to evaluate and check your answers numerically:
(a) lim
x →0+
(sina) ²
1
1
(b) lim ( - )
sin² x²
X](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Faf0432bf-c1b5-433e-bde2-eaf9b9deba63%2Fb7a0c6e3-ec25-4f01-b1ae-ffa698abb048%2Fq0vp0tk_processed.png&w=3840&q=75)
Transcribed Image Text:2) Let f be continuous on [a, b], differentiable on (a, b) and positive (i.e., > 0) for all
1
x = (a, b). Prove that there exists c = (a, b) such that-
f'(c) 1
f(c) a-c
=
+
(Hint:
b-c
(x − a) (x − b) ƒ (x) and use MVT for F(x) to show
consider the function F(x)
the existence of such c = (a, b).)
3) Use L'Hopital's Rule to evaluate and check your answers numerically:
(a) lim
x →0+
(sina) ²
1
1
(b) lim ( - )
sin² x²
X
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