Problem is in the attached picture, thank you. ### Mathematical Concepts and Problem-SolvingLet $ f $ be the continuous function defined on $ x \in [-1, 8] $ whose graph, consisting of two line segments, is shown at the right. Let $ g, h, $ and $ p $ be the functions defined by:\[g(x) = \sqrt{x^2 - x + 4}, \quad h(x) = 5e^x - 9 \sin x, \quad \text{and} \quad p(x) = -\frac{2}{x}\]#### Graph ExplanationThe graph of $ f $, labeled as "Graph of $ f $", consists of a V-shaped structure with an ascending line segment from $(0, 2)$ to $(3, 6)$, peaking at $(3, 6)$, and then descending to $(8, 2)$.#### Problem Statements**a.** The function $ k $ is defined by $ k(x) = f(x) \cdot g(x) $. Find $ k'(0) $. Simplify your answer completely.**b.** The function $ m $ is defined by $ m(x) = \frac{f(x)}{2h(x)} $. Find $ m'(0) $. Simplify your answer completely.**c.** Find the value of $ x $ for $ -1 < x < 2 $ such that $ f'(x) = p'(x) $.

Answered: Image /qna-images/answer/8f457272-53d9-4a4a-a2cc-b488d0adc2f2.jpg

Let f be the continuous function defined on x E [-1,8] whose graph, consisting of two line segments, is shown at the right. Let g, h, and p be the functions defined by g(x)=√x²-x+4, h(x) = 5ex − 9 sin x and 2 p(x) X 6- a. The function k is defined by k(x) = f(x) · g(x). Find k'(0). Simplify your answer completely. b. The function m defined by m(x): completely. c. Find the value of x for -1 < x < 2 such that f'(x) = p'(x). f(x) 2h(x)* O 4 Graph of f Find m'(0). Simplify your answer to too 8 X

Let f be the continuous function defined on x E [-1,8] whose graph, consisting of two line segments, is shown at the right. Let g, h, and p be the functions defined by g(x)=√x²-x+4, h(x) = 5ex − 9 sin x and 2 p(x) X 6- a. The function k is defined by k(x) = f(x) · g(x). Find k'(0). Simplify your answer completely. b. The function m defined by m(x): completely. c. Find the value of x for -1 < x < 2 such that f'(x) = p'(x). f(x) 2h(x)* O 4 Graph of f Find m'(0). Simplify your answer to too 8 X

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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Question

Problem is in the attached picture, thank you.

### Mathematical Concepts and Problem-Solving

Let $ f $ be the continuous function defined on $ x \in [-1, 8] $ whose graph, consisting of two line segments, is shown at the right. Let $ g, h, $ and $ p $ be the functions defined by:

\[
g(x) = \sqrt{x^2 - x + 4}, \quad h(x) = 5e^x - 9 \sin x, \quad \text{and} \quad p(x) = -\frac{2}{x}
\]

#### Graph Explanation

The graph of $ f $, labeled as "Graph of $ f $", consists of a V-shaped structure with an ascending line segment from $(0, 2)$ to $(3, 6)$, peaking at $(3, 6)$, and then descending to $(8, 2)$.

#### Problem Statements

**a.** The function $ k $ is defined by $ k(x) = f(x) \cdot g(x) $.
Find $ k'(0) $. Simplify your answer completely.

**b.** The function $ m $ is defined by $ m(x) = \frac{f(x)}{2h(x)} $.
Find $ m'(0) $. Simplify your answer completely.

**c.** Find the value of $ x $ for $ -1 < x < 2 $ such that $ f'(x) = p'(x) $.

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