Write a possible cubic polynomial inequality, in factored form, that gives the solutions x < - 3/2 and 2/5 < x < 7/4.

Question

Explain how you came up with the inequality (include a graph if you used a graphical method), then verify the solutions by including the sign table. Thank you very much for your help.

Answer 1

(x+3/2)(x-2/5)(x-7/4)<0

x+3/2<0

x<-3/2

or

x>2/5 and x<7/4

graphically the cubic is a U shape on the right and an upside down U shape on the left

It has 3 x intercepts, -3/2, 2/5 and 7/4

to the left of -3/2 the cubic is negative

(x+3/2)(x-2/5)(x-7/4)

or (2x+3)(5x-2)(4x-7) if you prefer integers

= (10x^2 +11x -6)(4x-7)

40x^3 +26x^2 +53x +42<0

or

x^3 +13x^2/20 +53x/40 +21/20<0 if you prefer a leading coefficient of 1

between 2/5 and 7/4 the cubic is negative

between -3/2 and 2/5, it's positive

greater than 7/4 it's positive

there is a local or relative maximum between x=-3/2 and x=2/5

a local or relative minimum between x=2/5 and x=7/4

an inflection point between the local max and local, where the U and upsidedown U are joined together. concave down to the left of the inflection point and cocave up to the right of the inflection point. the inflection point is where the 2nd derivative = 0. the local max and min are where the 1st derivative = 0.