Application of Derivatives

If the cost function of producing x units of an item is C(x) = 5x^2 + 20x + 100, what is the marginal cost when x = 10?

A rectangular field has length x meters and width (50 - x) meters. Find the value of x that maximizes the area of the field.

The revenue function for selling x units of a product is R(x) = 100x - 2x^2. Find the number of units to be sold to maximize revenue.

If y = x^3 - 6x^2 + 9x + 15, find the points where the function has local maxima or minima.

A manufacturer finds that the profit function P(x) = -5x^2 + 200x - 1500, where x is the number of units produced. How many units should be produced to maximize profit?

The speed of a particle at time t seconds is given by v(t) = 3t^2 - 12t + 9. Find the time when the particle is momentarily at rest.

The cost function of producing x units is C(x) = 50 + 4x + 0.1x^2. Find the production level where the marginal cost equals average cost.

The height of a projectile is given by h(t) = -5t^2 + 20t + 15 meters. Find the time at which the projectile reaches maximum height.

If the demand function is given by p = 100 - 2x, where p is price and x is quantity demanded, find the marginal revenue when x = 10.

Find the intervals where the function f(x) = x^3 - 3x^2 + 4 is increasing.