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use the given graph of f over the interval (0, 7) to find the following.

  1. The open interval on which f is increasing.
  2. The open interval on which f is decreasing.
  3. The open interval on which f is concave upward.
  4. The open interval on which f is concave downward.
  5. The coordinates of the points of inflection.

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Nestor Rutherford
Nestor RutherfordLv2
13 Mar 2020

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6. (14 pts) EXPLANATION IS NOT REQUIRED, AND NO PARTIAL CREDIT WILL BE GIVEN. The graph of f'(derivative of f) on the interval [-6, 7] is shown in the figure. y f(x) y = f'(x) -6 -5 -4 -3 2 L - 2 3 4 5 7 Use the given graph of f' to answer the following questions about f: (a) (2 pts) on what interval (or intervals) is f decreasing? (b) (3 pts) Find the critical points of f. (c) (2 pts) which critical points (or point) correspond to local maxima ? (d) (2 pts) which critical points (or point) correspond to neither local maxima nor local minima ? (e) (2 pts) on what interval (or intervals) is f concave up? (f) (3 pts) At what point (or points) does f have an inflection point ?

The graph of the derivative f โ€˜of a continuous function f is shown.

(a) On what intervals is f increasing? Decreasing?

(b) At what values of x does f have a local maximum? Local minimum?

(c) On what intervals is f concave upward? Concave downward?

(d) State the x-coordinate(s) of the point(s) of inflection

(e) Assuming that f (0) = 0, sketch a graph of f.

Assume that all of the functions are twice differentiable and the second derivatives are never 0.

If f and g are positive, increasing, concave upward functions on I, show that the product function f g is concave upward on I.

(b) Show that part (a) remains true if f and g are both decreasing.

(c) Suppose f is increasing and t is decreasing. Show, by giving three examples, that f g may be concave upward, concave downward, or linear. Why doesnโ€™t the argument in parts (a) and (b) work in this case?

(a)

Given:

Functions f and g are positive, increasing, concave upward functions on I.

Hence,

Now,

Hence, the product function is concave upward.

(b)

When both the functions are decreasing and concave upward

Hence, the product function is concave upward.

(c)

When f is increasing and g is decreasing

can either positive or negative on I, depending on values of on I..

Hence, the product function may be concave upward, concave downward or linear.

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