5. Given the function 𝑝(𝑥) = 𝑘 ∙ 𝑥^3 − 12𝑥^2 − 3750𝑥 + 180, where K=5

Question

Find (a) all critical points

(b) all extreme values

(c) all inflection points (give an ordered pair)

(d) identify a suitable interval that includes all these features, and

(e) provide a graph on this interval (not hand drawn; use desmos). Be sure to choose the graphing window appropriately so that the graph clearly shows the important features and there are no huge areas of empty space.

Answer 1

We will need the first two derivatives of p:

p(x) = 5x³ - 12x² - 3750x + 180

p'(x) = 15x² - 24x - 3750

p"(x) = 30x - 24

(a) The critical points of a differentiable functions, like p,

are those where the derivative is 0:

15x² - 24x - 3750 = 0

5x² - 8x - 1250 = 0

x = (8 +- sqrt(64 + 4 x 5 x 1250))/(2 x 5)

= (4 +- sqrt(16 + 5 x 1250))/5

= 16.63 and -15.03

(b) The problem statement does not limit us to any domain,

so we consider all real values x.

p(x) goes to -infinity when x goes to -infinity and

p(x) goes to +infinity when x goes to +infinity.

Therefore the function has no extreme values.

(c) An inflexion point occurs where the second derivative is 0:

30x - 24 = 0

x = 0.8

(d) The interesting features are at 16.63, -15.03, 0.8.

The interval [-20, +20] looks nice to me.

(e) This platform does not allow me to attach a file.