Let f and g be two functions whose ﬁrst and second order derivative functions are continuous,...

Let f and g be two functions whose ﬁrst and second order derivative functions are continuous, all deﬁned on R. What assumptions on f and g guarantee that the composite function f ◦g is concave?

Expert Solution

There are two methods to prove the theorem after the consideration of assumptions(the assumptions remain same for both cases).For your clarity the proofs have also been included.You may skip them if you want.

Colby Messinger answered 1 year ago

Let⇀F and⇀G be vector fields defined on R3 whose component functions have continuous partial derivatives. Furthermore,...

Let⇀F and⇀G be vector fields defined on R3 whose component functions have continuous partial derivatives. Furthermore, assume that ⇀∇×⇀F=⇀∇×⇀G.Show that there is a scalar function f such that ⇀G=⇀F+⇀∇f.

1) Let f(x) be a continuous, everywhere differentiable function and g(x) be its derivative. If f(c)...

1) Let f(x) be a continuous, everywhere differentiable function and g(x) be its derivative. If f(c) = n and g(c) = d, write the equation of the tangent line at x = c using only the variables y, x, c, n, and d. You may use point-slope or slope-intercept but do not introduce more variables. 2) Let f(x) be a continuous, everywhere differentiable function. What kind information does f'(x) provide regarding f(x)? 3) Let f(x) be a continuous, everywhere differentiable...

if f and g are the functions whose graphs are shown, let u(x) = f(x)g(x) and v(x) = f(x)/g(x).

if f and g are the functions whose graphs are shown, let u(x) = f(x)g(x) and v(x) = f(x)/g(x) (a) Find u'(1) (b) Find v'(5).

Let P(x) = F(x)G(x) and Q(x) = F(x)/G(x), where F and G are the functions whose graphs are shown.

Let P(x) = F(x)G(x) and Q(x) = F(x)/G(x), where F and G are the functions whose graphs are shown.(a) Find P ' (2)(b) Find Q ' (7)

Part A. If a function f has a derivative at x not. then f is continuous...

Part A. If a function f has a derivative at x not. then f is continuous at x not. (How do you get the converse?) Part B. 1) There exist an arbitrary x for all y (x+y=0). Is false but why? 2) For all x there exists a unique y (y=x^2) Is true but why? 3) For all x there exist a unique y (y^2=x) Is true but why?

Let p, q, g : R → R be continuous functions. Let L[y] := y'' +...

Let p, q, g : R → R be continuous functions. Let L[y] := y'' + py' + qy. (i) Explain what it means for a pair of functions y1 and y2 to be a fundamental solution set for the equation L[y] = 0. (ii) State a theorem detailing the general solution of the differential equation L[y] = g(t) in terms of solutions to this, and a related, equation.

Integral Let f:[a,b]→R and g:[a,b]→R be two bounded functions. Suppose f≤g on [a,b]. Use the information...

Integral Let f:[a,b]→R and g:[a,b]→R be two bounded functions. Suppose f≤g on [a,b]. Use the information to prove thatL(f)≤L(g)andU(f)≤U(g). Information: g : [0, 1] —> R be defined by if x=0, g(x)=1; if x=m/n (m and n are positive integer with no common factor), g(x)=1/n; if x doesn't belong to rational number, g(x)=0 g is discontinuous at every rational number in[0,1]. g is Riemann integrable on [0,1] based on the fact that Suppose h:[a,b]→R is continuous everywhere except at a...

1) The graph of the derivative f ' of a continuous function f is shown. (Assume...

1) The graph of the derivative f ' of a continuous function f is shown. (Assume the function f is defined only for 0 < x < ∞.) (a) On what interval(s) is f increasing? (2a) On what interval(s) is f decreasing? (b) At what value(s) of x does f have a local maximum? (2b) At what value(s) of x does f have a local minimum? x=? (c) On what interval(s) is f concave upward? (2c) On what interval(s) is...

Question 1 f is a function whose domain is (0,∞) and whose first derivative is f...

Question 1 f is a function whose domain is (0,∞) and whose first derivative is f ' (x) = (4 − x)x -3 for x > 0 a) Find the critical number(s) of f. Does f have a local minimum, a local maximum, or neither at its critical number(s)? Explain. b) On which interval(s) if f concave up, concave down? c) At which value(s) of x does f have a point of inflection, if any? Question 2 A rectangular box...

Let F and G~be two vector fields in R2 . Prove that if F~ and G~...

Let F and G~be two vector fields in R2 . Prove that if F~ and G~ are both conservative, then F~ +G~ is also conservative. Note: Give a mathematical proof, not just an example.

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