Suppose that f(x)=x^4+8x^3

Question

b.) Find all intercepts. x-intercept(s)=____ y-intercept(s)= 0

c.) Find all critical values of f. Critical value(s)=_______

d.) Use interval notation to indicate where f(x) is increasing or decreasing.

Increasing;_____ Decreasing: (-INF,-6)

e.) Find the x-coordinates of all local maxima and minima of f.

Local maxima at x=_____ Local minima at x=______

f.) Use interval notation to indicate where f(x) is concave up or down.

Concave up:_____ Concave down:_____

g.) Find all inflection points of f. Inflection point(s) at x=

h.) Find all asymptotes of f.

Horizontal asymptote(s): y= _____ Vertical asymptote(s): x=_____

Answer 1

y = x^4 - 8x^3 = x^3( x - 8)

x intercepts are zero and 8. zero is solution of multiplicity 3

y intercept at (0,0)

derivative: y' = 4x^3 - 24x^2 = 4x^2 ( x - 6)

critical values are x=0 of multiplicity 2, and x=6

again, that's (0,0) and (6,-432)

2nd derivative: y'' = 12x^2 - 48x = 12x( x- 4)

inflection points x=0 and x=4

(0,0) and (4, -256)