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 with a square base has a surface area of 150 cm² , obtained by adding the areas all six sides of the box. Find the dimensions of the box with the largest possible volume.

Expert Solution

We have to eliminate x = 0 because the denominator will also become 0.

milcah answered 2 years ago

Question 1 Let f be a function for which the first derivative is f ' (x)...

Question 1 Let f be a function for which the first derivative is f ' (x) = 2x 2 - 5 / x2 for x > 0, f(1) = 7 and f(5) = 11. Show work for all question. a). Show that f satisfies the hypotheses of the Mean Value Theorem on [1, 5] b)Find the value(s) of c on (1, 5) that satisfyies the conclusion of the Mean Value Theorem. Question 2 Let f(x) = x 3 − 3x...

Let f be a differentiable function on the interval [0, 2π] with derivative f' . Show...

Let f be a differentiable function on the interval [0, 2π] with derivative f' . Show that there exists a point c ∈ (0, 2π) such that cos(c)f(c) + sin(c)f'(c) = 2 sin(c).

Consider the function f(x)f(x) whose second derivative is f''(x)=5x+10sin(x)f′′(x)=5x+10sin(x). If f(0)=4f(0)=4 and f'(0)=4f′(0)=4, what is f(5)f(5)?....

Consider the function f(x)f(x) whose second derivative is f''(x)=5x+10sin(x)f′′(x)=5x+10sin(x). If f(0)=4f(0)=4 and f'(0)=4f′(0)=4, what is f(5)f(5)?. show work

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...

Use the given function, its first derivative, and its second derivative to answer the following: f(x)=(1/3)x^3...

Use the given function, its first derivative, and its second derivative to answer the following: f(x)=(1/3)x^3 - (1/2)x^2 - 6x + 5 f'(x)= x^2 - x - 6 = (x+2)(x-3) f''(x)= 2x - 1 a) What are the intervals of increase and the intervals of decrease b) Identify local min and max points c) What are the intervals where the function is concave up, concave down and identify the inflection points

Let f(x, y) be a function such that f(0, 0) = 1, f(0, 1) = 2,...

Let f(x, y) be a function such that f(0, 0) = 1, f(0, 1) = 2, f(1, 0) = 3, f(1, 1) = 5, f(2, 0) = 5, f(2, 1) = 10. Determine the Lagrange interpolation F(x, y) that interpolates the above data. Use Lagrangian bi-variate interpolation to solve this and also show the working steps.

Is it important to know the domain of a logarithmic function before finding the derivative? Why?

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?

Consider the function f(x)= x3 x2 − 1 Express the domain of the function in interval...

Consider the function f(x)= x3 x2 − 1 Express the domain of the function in interval notation: Find the y-intercept: y= . Find all the x-intercepts (enter your answer as a comma-separated list): x= . On which intervals is the function positive? On which intervals is the function negative? Does f have any symmetries? f is even;f is odd; f is periodic;None of the above. Find all the asymptotes of f (enter your answers as equations): Vertical asymptote (left): ; Vertical...

Given the first derivative of a function is f'(×)=2x^2-8x+10 , find the intervals of increasing and...

Given the first derivative of a function is f'(×)=2x^2-8x+10 , find the intervals of increasing and decreasing behavior of the function.

Question