Hello, can you please help me with this problem. Thank you. Please show all your work. Find all numbers b such that the average value of f(x) = 6 + 10x -9x² on the interval [0, b] is equal to 7.

Answered: Find all numbers b such that the…

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

Hello, can you please help me with this problem. Thank you. Please show all your work.

Find all numbers b such that the average value of f(x) = 6 + 10x -
9x² on the interval [0, b] is equal to 7.

Expert Solution

Step 1: Given Information and formula used :

The function $f (x) = 6 + 10 x - 9 x^{2}$ on the interval $[0, b]$ and the average value of the function is $7$ .

The objective is to find all number $b$ .

Formula Used :

(1) The average value of the function $f (x)$ in the interval $[a, b]$ is $f_{a v e} = \frac{1}{b - a} \int_{a}^{b} f (x) d x$ .

(2) Power rule of integration is $\int x^{n} d x = \frac{x^{n + 1}}{n + 1} + c$ where $c$ is the integrating constant.

(3) The fundamental theorem of calculus states that If there is an antiderivative $F$ of the function in the interval $[a, b]$ , then the definite integral of the function is the difference between the values of $F$ that is $F (b) - F (a) = \int_{a}^{b} f (x) d x$ .