A. At what labeled point(s) is f(x) increasing. Justify your answer. B. At what labeled point(s) is f(x) concave u[p. Justify your answer. C. For what values of x does the graph of f have a horizontal tangent? Justify your answer. D. Let h be a function defined for all x #0 such that h(2) = -4 and the derivative of his given by h'(x)=x²-2 for all x 0. Write an equation for the line tangent to the graph of h at x = 2.
A. At what labeled point(s) is f(x) increasing. Justify your answer. B. At what labeled point(s) is f(x) concave u[p. Justify your answer. C. For what values of x does the graph of f have a horizontal tangent? Justify your answer. D. Let h be a function defined for all x #0 such that h(2) = -4 and the derivative of his given by h'(x)=x²-2 for all x 0. Write an equation for the line tangent to the graph of h at x = 2.
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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Transcribed Image Text:**Transcription for Educational Use**
**Part 1**
The figure above shows the graph of \( f' \), the derivative of a function \( f \).
**Questions:**
A. At what labeled point(s) is \( f(x) \) increasing? Justify your answer.
B. At what labeled point(s) is \( f(x) \) concave up? Justify your answer.
C. For what values of \( x \) does the graph of \( f \) have a horizontal tangent? Justify your answer.
D. Let \( h \) be a function defined for all \( x \neq 0 \) such that \( h(2) = -4 \) and the derivative of \( h \) is given by \( h'(x) = \frac{x^2 - 2}{x} \) for all \( x \neq 0 \). Write an equation for the line tangent to the graph of \( h \) at \( x = 2 \).
**Graph Description:**
The graph shows a curve depicting \( f' \). It begins below the x-axis, increases to a peak above the axis at point B, descends through zero to a minimum past x=2, and then rises again. Points A, B, C, and D are labeled on the curve, presumably at distinct values of \( x \).
- Point A is approximately at (-4, -8).
- Point B is at (-2, 10).
- Point C is slightly above 0 on the x-axis.
- Point D is around (4, -4).
This content could be used in a calculus course to discuss properties of derivatives and their relation to the behavior of function graphs, such as increasing/decreasing intervals, concavity, and tangent lines.
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