Question 7 14 Let f be a twice differentiable function, and let f(6) = 7, f'(6)=0,...
Let f be a twice differentiable function on an open interval (a, b). Which statements regarding the second derivative and concavity are true? If f"(c) is positive, then the graph of f has a local maximum at x = c. The concavity of a graph changes at an inflection point. If f is increasing, then the graph of f is concave down. The graph of f has a local minimum at x = c if f"(c) = 0. The graph of f is concave up if...
Question 6 (1 point) Suppose a function f(x) is differentiable everywhere and has a local minimum at x=c. If f(x)<O when x<c, and f'(x)>0 when x>c, then by the Global Interval Method we know x=c is O a local maximum an absolute maximum a local minimum an absolute minimum
Use the table below for twice- differentiable (i.e. it can have its derivative taken twice without incident) y =h(x). X -2 1 2 3 6 6 7 6 3 10 -1 h'(x) 3 Answer both questions below, justifying the answers: Where does y=h(x)have a relative minimum over (-2,10)? Where does y=h(x)have an inflection point over (-2,10)?
2. Let g(x) In f(x) where f(x) is a twice differentiable positive function on (0, o) such that f(x + 1) = x f(x) Then for N 1, 2, 3 find g" N+ 2 2. Let g(x) In f(x) where f(x) is a twice differentiable positive function on (0, o) such that f(x + 1) = x f(x) Then for N 1, 2, 3 find g" N+ 2
Suppose that f is twice differentiable function where f(0)=f(1)=0. Prove that strategy Suppose that f is a twice differentiable function where f(0) = f(1) = 0. 1 Prove that f f"(x)f (x) dx a. Using part a, show that if f"(x) = wf (x) for some constant w, then w 0. Can you think of a function that satisfies these conditions for some nonzero w? b. strategy Suppose that f is a twice differentiable function where f(0) = f(1) =...
7. [23] Given the following function:: f(x)-x-4x +6 (a) Find all of the critical points of this function. Show your work. (b) Characterize each of the critical points as a local maximum, a local minimum or neither. Show your work. (c) Find all of the inflection points of this function (verify that it/they are indeed inflection points). (d) On what interval(s) is this function both decreasing and concave down? on the interval -15xs1. Show (e)Find the global maximum and minimum...
Question 11 10 pts The derivative f'(2) of an unknown function f(x) has been determined as f'(x) = (x - 2)(+3)2. Use this derivative to find the intervals where the original function f is increasing/decreasing. Then find the x-values that correspond to any relative maximums or relative minimums of the original unknown function f(x). O no relative maximum; relative minimum at x=2 relative maximum at x=-3; no relative minimum O relative maximum at x=2; relative minimum at x=-3 relative maximum...
Suppose that f is a twice differentiable function and that its second partial derivatives are continuous. Let h(t) = f(x(t), y(t)) where x = 2e and y = 2t. Suppose that f:(2,0) = 4, fy(2,0) = 3, fx=(2,0) = 2, fyy(2,0) = 3, and fxy(2,0) = 2. Find out that when t=0.
7) Let O S Rn be open and suppose f : O → R is differentiable on O. Suppose has a local maximum or minimum at zo E O. Prove that f'(zo) = 0. 7) Let O S Rn be open and suppose f : O → R is differentiable on O. Suppose has a local maximum or minimum at zo E O. Prove that f'(zo) = 0.
15-16 The graph of the derivative f' 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. 15. y A y = f'(x) --2 0 2 6 8 x -2