This is an ungraded practice prompt. ### Twice-Differentiable Functions and Analysis#### Table of Values:The table provided lists values for $ x $, $ f(x) $, $ f'(x) $, $ g(x) $, and $ g'(x) $ for selected values of $ x $. | $ x $ | $ f(x) $ | $ f'(x) $ | $ g(x) $ | $ g'(x) $ ||--------|------------|-------------|------------|-------------|| 1 | 6 | -2 | 1 | 10 || 2 | 4 | -8 | 12 | -9 || 3 | 1 | -3 | 3 | 2 || 4 | -1 | 2 | -8 | -4 |#### Problem Analysis:The functions $ f $ and $ g $ are twice-differentiable for all real numbers and exhibit continuous second derivatives. The graph of $ f $ is strictly decreasing. We are given values of $ f $, $ f' $, $ g $, and $ g' $ for specific $ x $ values. Define $ h $ as $ h(x) = f(x) \cdot g(x) $.#### Questions:a) **Explain why there must be a value $ a $ for $ 1 < a < 4 $ such that $ h(a) = 7 $****Answer:** By the Intermediate Value Theorem, if $ h(x) $ is continuous and takes different values at two points, then it must take every value between them at some point in the interval. Given $ h(1) = f(1) \cdot g(1) = 6 \cdot 1 = 6 $ and $ h(4) = f(4) \cdot g(4) = (-1) \cdot (-8) = 8 $, and 7 is between 6 and 8, there exists some $ a $ in $ (1, 4) $ such that $ h(a) = 7 $.b) **Find $ h'(3) $****Answer:** The derivative of $ h(x) $ is \( h

According to guidelines I solve first 3 subparts so kindly request you to repost the remaining…

Answered: Explain why there must be a value a for…

Explain why there must be a value a for 1 < a < 4 such that h(a) = 7. Find h '(3). Let f¹ be the inverse function of f. Write an equation for the line tangent to the graph of y = f'(x) at x = 4.

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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