In Exercises 29–32 the graph of the second derivative, f ″ ( x ) , is given. Determine the x-coordinates of all points of inflection of f ( x ) , if any. (Assume that f ( x ) is defined and continuous everywhere in [ − 3 , 3 ] .) [ HINT: Remember that a point of inflection of f corresponds to a point at which f ″ schanges sign, from positive to negative or vice versa. This could be a point where its graph crosses the x -axis or a point where its graph is broken: positive on one side of the break and negative on the other.]
In Exercises 29–32 the graph of the second derivative, f ″ ( x ) , is given. Determine the x-coordinates of all points of inflection of f ( x ) , if any. (Assume that f ( x ) is defined and continuous everywhere in [ − 3 , 3 ] .) [ HINT: Remember that a point of inflection of f corresponds to a point at which f ″ schanges sign, from positive to negative or vice versa. This could be a point where its graph crosses the x -axis or a point where its graph is broken: positive on one side of the break and negative on the other.]
Solution Summary: The author explains the x -coordinates of the point of inflection of a function.
In Exercises 29–32 the graph of the second derivative,
f
″
(
x
)
, is given. Determine the x-coordinates of all points of inflection of
f
(
x
)
, if any. (Assume that
f
(
x
)
is defined and continuous everywhere in
[
−
3
,
3
]
.) [HINT: Remember that a point of inflection of f corresponds to a point at which
f
″
schanges sign, from positive to negative or vice versa. This could be a point where its graph crosses the x-axis or a point where its graph is broken: positive on one side of the break and negative on the other.]
2. Suppose that f(x) is a function continuous for every value of x whose first derivative is f'(x) = 2(1-x)/1+x^2 and f"(x)= 4x(x^2-3)/ (1+x^2)^2 Further, assume that it is known that f has a horizontal asymptote at y = 0. and
a. Determine all critical points of f.
Consider a function y =f(t) with domain (0. 1, 4. 2).Use the graph of the derivative function, y=f'(t) ,given below to write down the t -value of the local maximum of f. If the function has no local maximum, explain why not
Answer the following questions about the function whose derivative is
f′(x)=(x−4)(x+8)(x−6).
a. What are the critical points of f?
b. On what open intervals is f increasing or decreasing? [In interval notation for b]
c. At what points, if any, does f assume local maximum and minimum values?
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Differential Equation | MIT 18.01SC Single Variable Calculus, Fall 2010; Author: MIT OpenCourseWare;https://www.youtube.com/watch?v=HaOHUfymsuk;License: Standard YouTube License, CC-BY