Derivatives (229 problems)

Apply the law of logarithms to simplify the function

Then, find its derivative.

Use logarithmic differentiation to find

Given f(x)= -2x³+6x².

a) Find the intervals over which f(x) is increasing.

b) Find the maximum and minimum values of f(x) on [-1,1].

c) Where is f(x) concave up?

For the following function, and the given interval, compute all the absolute extrema.

Determine the critical points for.

If the revenue is

and the cost is

Compute the marginal profit and find the point that maximizes the profit.

What is the maximum possible volume of a right circular cylinder with a total surface area of(including the top and bottom)

Given, on

(a) Find the critical numbers of the first-order

(b) Find the intervals of increase and decrease. Show work that supports conclusion

(c) Give the coordinates of the local maximum, if any

(d) Give the coordinates of the local minimum, if any

(e) Find the intervals on which f(x) is concave up and the intervals on which it is concave down. Show the work. Find the coordinates of the inflection point, if there is any

420 meters of fencing must be used to enclose a rectangular area on all sides and also divide the rectangular region into two smaller rectangles. The larger of these rectangles must have a length that is three times the length of the smaller rectangle. Determine the overall dimensions of the rectangular area that maximizes the total area enclosing the two smaller rectangular regions and determine the dimensions of each inner rectangular area.

Use the second derivative test to find the x-coordinates of the relative maximum and the relative minimum points of the following function.

Let f(x) = (x+18)(x-9)² – 5

a) Find any local extrema. Say at what value of x they occur, and say whether each is a maximum or a minimum.

b) Find the absolute minimum of f(x) on (0,∞).

Does the functiontake on a maximum value on the interval (0,1)? Explain

For speeds between 35 and 65 miles per hour, a diesel truck gets 480/x miles per gallon when driven at a constant speed of x miles per hour. Diesel gasoline costs $2.80 per gallon and the driver is paid $14 per hour. What is the most economical constant speed between 35 and 65 miles per hour at which the truck should be driven?

What is the maximum possible area of a rectangle inscribed in the ellipse, with the sides of the rectangle parallel to the coordinate axes?

Use the second derivative test to find the x-coordinates of the relative maximum and the relative minimum points of the following function.

Give an example of a function that has a critical number that is not a maximum or minimum of the function. And tell what the number is.

Determine the dimensions of a rectangular solid with a square base with maximum volume if its surface area is 337.5 square centimeters. Show all work.