### HW 8.5 #1:**Problem Statement:**Suppose $ f $ and $ g $ are continuous on the interval $[a, b]$ and that $ f(a) < g(a) $ and $ f(b) > g(b) $. Prove that there is a real number $ x_0 \in (a, b) $ such that $ f(x_0) = g(x_0) $.**Hint:**Define a function $ h(x) = f(x) - g(x) $ and apply the Intermediate Value Theorem.### Detailed Explanation:The problem requires proving that two continuous functions, $ f $ and $ g $, intersect at some point within the interval $(a, b)$. This can be approached using the concept of the Intermediate Value Theorem (IVT) which is a fundamental theorem in calculus.**Steps:**1. **Define a New Function:** - Define $ h(x) = f(x) - g(x) $. - Since $ f $ and $ g $ are continuous, and the difference of continuous functions is continuous, $ h $ is continuous on $[a, b]$.2. **Evaluate at the Endpoints:** - $ h(a) = f(a) - g(a) $. Given $ f(a) < g(a) $, we have $ h(a) < 0 $. - $ h(b) = f(b) - g(b) $. Given $ f(b) > g(b) $, we have $ h(b) > 0 $.3. **Apply the Intermediate Value Theorem:** - $ h(x) $ is continuous, and since it changes signs from $ a $ to $ b $ (i.e., $ h(a) < 0 $ to $ h(b) > 0 $), by the IVT, there exists some $ x_0 \in (a, b) $ where $ h(x_0) = 0 $.4. **Conclusion:** - Thus, $ h(x_0) = 0 \implies f(x_0) - g(x_0) = 0 \implies f(x_0) = g(x_0) $.This proves there is a real number $ x_0 $ in

Name: ### HW 8.5 #1:**Problem Statement:**Suppose $ f $ and $ g $ are c
Uploaded: 2024-03-20T06:46:42.000Z
Channel: Bartleby
Description: ### HW 8.5 #1:**Problem Statement:**Suppose $ f $ and $ g $ are continuous on the interval $[a, b]$ and that $ f(a) g(b) $. Prove that there is a real number $ x_0 \in (a, b) $ such that $ f(x_0) = g(x_0) $.**Hint

Answered: Image /qna-images/answer/b7ac29a9-99e7-454a-8547-c2f790b2afa0.jpg

Math

Advanced Math

HW 8.5 #1: Suppose f and g are continuous on the interval [a, b] and that f(a) < g(a) and f(b) > g(b). Prove that there is a real number ro E (a, b) such that (Ox)6 = (0x)S S(r) – g(x) and apply the Intermediate Value (HINT: Define a function h(r) Theorem.)

HW 8.5 #1: Suppose f and g are continuous on the interval [a, b] and that f(a) < g(a) and f(b) > g(b). Prove that there is a real number ro E (a, b) such that (Ox)6 = (0x)S S(r) – g(x) and apply the Intermediate Value (HINT: Define a function h(r) Theorem.)

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Related questions

Q: Prove the following statement: Assume f is a continuous function on [a, b], k is any real number |…

Q: 6 language of a function f being continous at a point zo € dom(ƒ). Let f(x) = 2x² +1. Using the e- &…

Q: Exercise 1. Let f(x) be the function 2х — 1 if x 2. - (a) Find the value of c that makes f(x)…

A: Given f(x)=2x-1if x<-1cxif -1≤x≤2x-2if x>2

Q: Definition A function f(x) is continuous from the right at a number a if... And f(x) is continuous…

Q: Prove that if f is a continuous function on a closed interval [a, b], then

A: We have to prove the given inequality for a continuous function on a closed interval [a,b].

Q: Let f (x) = x + x3 + 2x 1. The function f is increasing on the following interval(s): (а, b) O [a,…

Q: Show that the function f(x)=x4 + 6x +1 has exactly one zero in the interval 1-1 01 Which theorem can…

Q: アー)。 The xy-plane above shows the graph of a continuous function f that is defined on interval [-2,…

A: We use the graph of f(x) and the definition of the function g(x) to answer the given question.

Q: : Let f [a, b] → R be a continuous and one-to-one function, where a < b are real numbers. Use the…

A: Introduction: There must be at least one "c" such that c∈(a,b) (or a<c<b) and f(c) = L. if…

Q: Determine L[f]if 0, f(1) = {1- 1, 0 2. 1,

A: Given that, A function f(t), t ≥ 0, its Laplace transform F(s) = L{f(t)} is defined as

Q: If ƒ : R → R and g : R → R are both one-to-one, is f + g one-to-one? If f and g are both onto, is f…

A: As per the guideline I’m supposed to solve only first question from the given questions. And the…

Q: (6 pts) Let g: A →→ B and f: BC be two functions. Prove or disprove the followings: (a) if f and g…

A: The objective of this question is to prove or disprove two statements about the composition of two…

Q: Provide an example of function f: R → R which is continuous at exactly one point and discontinuous…

Q: 1. Define function f : Z+ → Z such that f(n) = /n. (a) Is f one-to-one? Justify (prove) your answer.

Q: Determine values of a, b that belong to Real Numbers so that attached function be continuous in all…

A: Step 1: Step 2: Step 3: Step 4:

Q: Theorem 18.1 states that "Let f be a continuous real-valued function on a closed interval [a, b].…

Q: Let f: IR be a function. (a) Prove that if f is differentiable at some c I for which f(c) ‡ 0, then…

Q: O Find the largest value of A such that the function f(x) = x + 1x? - 4x + 6 is decreasing for all x…

Q: 4. (a) (b) Let f : X → Y and g: Y→ Z be functions. Prove the following statements. If gof is…

Q: Prove or disprove: f : Z30 → Z15 given by f([x]) = [x²] is a well-defined function.

Q: Show that the function r(0) = sec 0- + 5 has exactly one zero in the interval 0,5 every point over…

A: Given:- Show that the function r(θ) = sec θ-1θ3+5- has exactly one zero in the…

Topic Video

Question

### HW 8.5 #1:

**Problem Statement:**
Suppose $ f $ and $ g $ are continuous on the interval $[a, b]$ and that $ f(a) < g(a) $ and $ f(b) > g(b) $. Prove that there is a real number $ x_0 \in (a, b) $ such that $ f(x_0) = g(x_0) $.

**Hint:**
Define a function $ h(x) = f(x) - g(x) $ and apply the Intermediate Value Theorem.

### Detailed Explanation:
The problem requires proving that two continuous functions, $ f $ and $ g $, intersect at some point within the interval $(a, b)$. This can be approached using the concept of the Intermediate Value Theorem (IVT) which is a fundamental theorem in calculus.

**Steps:**
1. **Define a New Function:**
- Define $ h(x) = f(x) - g(x) $.
- Since $ f $ and $ g $ are continuous, and the difference of continuous functions is continuous, $ h $ is continuous on $[a, b]$.

2. **Evaluate at the Endpoints:**
- $ h(a) = f(a) - g(a) $. Given $ f(a) < g(a) $, we have $ h(a) < 0 $.
- $ h(b) = f(b) - g(b) $. Given $ f(b) > g(b) $, we have $ h(b) > 0 $.

3. **Apply the Intermediate Value Theorem:**
- $ h(x) $ is continuous, and since it changes signs from $ a $ to $ b $ (i.e., $ h(a) < 0 $ to $ h(b) > 0 $), by the IVT, there exists some $ x_0 \in (a, b) $ where $ h(x_0) = 0 $.

4. **Conclusion:**
- Thus, $ h(x_0) = 0 \implies f(x_0) - g(x_0) = 0 \implies f(x_0) = g(x_0) $.

This proves there is a real number $ x_0 $ in

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

SEE SOLUTION Check out a sample Q&A here

Step 1

VIEW

Step 2

VIEW

Step by step

Solved in 2 steps with 2 images

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

$Chain Rule$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.

Similar questions

D. Let function g: R → R be defined by g(x) - 2x + 1 when x 1; Find a real number c for which function g will be continuous at every point when x > 1. 5- cx in its domain. Why does your choice of c satisfy this criterion?
How would you answer #1c?
Let f be a function defined by x2 - c, if x 5 If f is continuous for all real numbers x, what is the value of c? Oc = Oc = 0 Ос- Oc = 2 O None of the above
Prove that the function f: R → R given by f(x) = 3x³ – 2x is continuous at 1 using the a. sequential definition, b. e - 8 definition.