Application of Derivatives

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

The revenue function for a product is R(x) = 50x - 0.5x^2, where x is the number of units sold. Find the number of units that maximizes revenue.

A rectangle is inscribed under the parabola y = 16 - x^2 and above the x-axis. What is the maximum area of such a rectangle?

If s(t) = t^3 - 6t^2 + 9t represents the position of a particle at time t, at what time is the velocity zero?

Find the points on the curve y = x^3 - 3x where the tangent is horizontal.

A spherical balloon is being inflated so that its radius increases at a rate of 2 cm/s. At what rate is the volume increasing when the radius is 5 cm? (Volume V = (4/3)πr^3)

The profit function of a company is P(x) = -2x^3 + 15x^2 + 36x - 10, where x is the number of units produced. Find the number of units that will maximize profit.

If the function f(x) = x^4 - 4x^3 + 6x^2 - 4x + 1, find the minimum value of f(x).

A wire of length 24 cm is cut into two pieces. One piece is bent into a square and the other into a circle. How should the wire be cut to minimize the total area?

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