Question

Let f(x) be a continuous, everywhere differentiable function. What kind information does f'(x) provide regarding f(x)?

Let f(x) be a continuous, everywhere differentiable function. What kind information does f''(x) provide regarding f(x)?

Let f(x) be a continuous, everywhere differentiable function. What kind information does f''(x) provide regarding f'(x)?

Let h(x) be a continuous function such that h(a) = m and h'(a) = 0. Is there enough evidence to conclude the point (a, m) must be a maximum or a minimum? Explain.

Write a function k(x) such that k(5) = -3, k'(5) = 0, and k''(5) < 0. Algebraically show your function k(x) does possess these attributes.

Answer #1

4a). Let g be continuous at x = 0. Show that f(x) = xg(x) is
differentiable at x = 0 and f'(0) = g(0).
4b). Let f : (a,b) to R and p in (a,b). You may assume that f is
differentiable on (a,b) and f ' is continuous at p. Show that f'(p)
> 0 then there is delta > 0, such that f is strictly
increasing on D(p,delta). Conclude that on D(p,delta) the function
f has a differentiable...

give an example of a continuous and differentiable function at a
point x = a, with a null derivative at this point, f '(a) = 0, and
that f has neither a maximum nor a minimum at x = a

generate a continuous and differentiable function with the
following properties: f(x) is decreasing at x=-5, local minimum is
x=-3, local maximum is x=3

Let f be continuously differentiable function on the Reals with
the following characteristics: - f(x) is increasing from intervals
(0,2) and (4,5) and decreasing everywhere else - f(x) > -1 on
the interval (1,3) and f(x) < -1 everywhere else Suppose g(x) =
2f(x) + (f(x))^2. On which interval(s) is g(x) increasing?

Does the function satisfy the hypotheses of the Mean Value
Theorem on the given interval?
f(x) = x3 + x − 9, [0, 2]
Yes, f is continuous on [0, 2] and differentiable on
(0, 2) since polynomials are continuous and differentiable on .No,
f is not continuous on [0, 2]. No,
f is continuous on [0, 2] but not differentiable on (0,
2).Yes, it does not matter if f is continuous or
differentiable; every function satisfies the Mean Value
Theorem.There is...

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^ t and y = 2t. Suppose that
fx(2, 0) = 1, fy(2, 0) = 3, fxx(2, 0) = 4, fyy(2, 0) = 1, and
fxy(2, 0) = 4. Find d ^2h/ dt ^2 when t = 0.

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 = 3e ^t and y = 2t. Suppose that
fx(3, 0) = 2, fy(3, 0) = 1, fxx(3, 0) = 3, fyy(3, 0) = 2, and
fxy(3, 0) = 1. Find d 2h dt 2 when t = 0.

An everywhere continuous function f satisfies f''(x)=
x2 -4x -5. Find the intervals where f is concave up and
concave down.

Does the function satisfy the hypotheses of the Mean Value
Theorem on the given interval? f(x) = 3x2 + 3x + 6, [−1, 1]
No, f is continuous on [−1, 1] but not differentiable
on (−1, 1).
There is not enough information to verify if this function
satisfies the Mean Value Theorem.
Yes, it does not matter if f is continuous or
differentiable; every function satisfies the Mean Value
Theorem.
No, f is not continuous on [−1, 1].Yes, f is...

1. Does the function satisfy the hypotheses of the Mean Value
Theorem on the given interval?
f(x) = x3 + x − 5, [0, 2]
a) No, f is continuous on [0, 2] but not differentiable
on (0, 2).
b) Yes, it does not matter if f is continuous or
differentiable; every function satisfies the Mean Value
Theorem.
c) There is not enough information to verify if this function
satisfies the Mean Value Theorem.
d) Yes, f is continuous on [0,...

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