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 balloon is rising vertically at a rate of 5 m/s. A boy is cycling on a straight road at 12 m/s. When the boy is 13 m away from the point on the road beneath the balloon, the balloon is 20 m high. What is the rate at which the distance between the boy and the balloon is changing?

Find the points on the curve y = x^3 - 3x^2 + 4 where the tangent is horizontal.

A manufacturer finds that the profit function P(x) in thousands of rupees for selling x units is P(x) = -2x^3 + 30x^2 + 150x - 200. Find the number of units to be sold to maximize profit.

If the displacement of a particle is given by s(t) = t^4 - 4t^3 + 6t^2, find the acceleration at t = 2 seconds.

The revenue function R(x) = 50x - 0.5x^2 represents the revenue (in thousand rupees) from selling x units of a product. Find the number of units that maximize the revenue.

A function f(x) is such that f'(x) = 6x^2 - 12x + 6. Find the intervals where f(x) is increasing.

Find the maximum area of a rectangle inscribed under the curve y = 12 - x^2 and above the x-axis.

The cost function is C(x) = 100 + 20x - 0.05x^2. Find the level of production x that minimizes the average cost.

If the volume of a cube is increasing at a rate of 150 cm³/s, find the rate at which the surface area is increasing when the edge length is 5 cm.