4. a. Let f(x) = ax²+3x+y be a quadratic function with a, 3, 7 € R. Consider the closed interval [a, b] which has midpoint c = ab. Prove that the slope of the secant line join- ing (a, f(a)) to (b, f(b)) is equal to the slope of the tangent line to f(x) at the point c. b. What you did in part a. was prove that the midpoint c is the same as the point c specified in the Mean Value Theorem applied to f(x) on the interval [a, b]. This is a special property of quadratic functions. It does not work in general. Find a counter example to demonstrate this. Your counterexample should specify both the function and the interval you are looking at.
4. a. Let f(x) = ax²+3x+y be a quadratic function with a, 3, 7 € R. Consider the closed interval [a, b] which has midpoint c = ab. Prove that the slope of the secant line join- ing (a, f(a)) to (b, f(b)) is equal to the slope of the tangent line to f(x) at the point c. b. What you did in part a. was prove that the midpoint c is the same as the point c specified in the Mean Value Theorem applied to f(x) on the interval [a, b]. This is a special property of quadratic functions. It does not work in general. Find a counter example to demonstrate this. Your counterexample should specify both the function and the interval you are looking at.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![4.
a. Let f(x) = ax²+3x+y be a quadratic function with a, 3,7 € R. Consider the closed
interval [a, b] which has midpoint c = a+b. Prove that the slope of the secant line join-
ing (a, f(a)) to (b, f(b)) is equal to the slope of the tangent line to f(x) at the point c.
b. What you did in part a. was prove that the midpoint c is the same as the point c
specified in the Mean Value Theorem applied to f(x) on the interval [a, b]. This is a
special property of quadratic functions. It does not work in general. Find a counter
example to demonstrate this. Your counterexample should specify both the function
and the interval you are looking at.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdac761a6-ba1c-4b50-93cd-f82ea0746880%2F83865cd0-7838-424e-8b4f-bc61821bfa16%2F5ph5mvn_processed.jpeg&w=3840&q=75)
Transcribed Image Text:4.
a. Let f(x) = ax²+3x+y be a quadratic function with a, 3,7 € R. Consider the closed
interval [a, b] which has midpoint c = a+b. Prove that the slope of the secant line join-
ing (a, f(a)) to (b, f(b)) is equal to the slope of the tangent line to f(x) at the point c.
b. What you did in part a. was prove that the midpoint c is the same as the point c
specified in the Mean Value Theorem applied to f(x) on the interval [a, b]. This is a
special property of quadratic functions. It does not work in general. Find a counter
example to demonstrate this. Your counterexample should specify both the function
and the interval you are looking at.
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