Topic 5.6: Area Under a Curve Questions

At a stationary point of f, which of the following is always true?

A stationary point has f'(x) changes from positive to negative. What type is it?

Find the stationary point of f(x) = x² − 4x + 7 and determine its nature.

Where is the y-coordinate of the stationary point of f(x) = 2x² − 8x + 3?

Find the coordinates of the stationary point of f(x) = −x² + 6x − 8 and state its nature.

f(x) = x² + 4x − 12. Find the minimum value of f.

The height of a ball is h(t) = −5t² + 20t + 2 metres. Find the maximum height.

p(x) = x³ − 3x. Find the local maximum and local minimum values.

Explain how a sign diagram of f'(x) can classify a stationary point. Give an example.

f(x) = x³. Which statement about x = 0 is correct?

Find and classify all stationary points of g(x) = 2x³ − 9x² + 12x − 4.

A function f has f'(x) = 3x² − 6x.

A curve has equation y = x³ − 6x² + 9x + 1.

A farmer encloses a rectangular paddock against a straight river, so only three sides need fencing: two equal widths w metres and one length parallel to the river. She has 120 m of fencing in total.

(a) Show that the enclosed area is A = 120w − 2w².

(b) Find the value of w that gives the maximum area, and state that maximum area.

f(x) = ax³ + bx² has a stationary point at (2, 4). Find a and b.

g(x) = x⁴ − 4x². Find all stationary points and classify them.

The running cost of a delivery van, in dollars per 100 km, is modelled by C(v) = 0.5v² + 1000/v, where v is the average speed in km h⁻¹ (v > 0).

(a) Find C′(v).

(b) Find the speed v that minimises the running cost, and find this minimum cost.