smc-750-50-Cubics

Algebra, MET2 2011 VCAA 18 MC

The equation `x^3 - 9x^2 + 15x + w = 0` has only one solution for `x` when

A. `−7 < w < 25`

B. `w <= −7`

C. `w >=25`

D. `w < −7` or `w > 25`

E. `w > 1`

Show Answers Only

`=> D`

Show Worked Solution

`y`	`=x^3 – 9x^2 + 15x + w`
`y′`	`=3x^2 – 18x + 15`
	`=3(x-5)(x-1)`

`text(Stationary points at)\ \ x=5 and 1.`

`text{Sketch the graph}\ \ (w=0),`

`text(By inspection, one solution occurs when)`

`w > 25\ \ text{(shift curve up > 25), or}`

`w < −7\ \ text{(shift curve down > 7)`

`=> D`

Graphs, MET2 2015 VCAA 17 MC

A graph with rule `f(x) = x^3 - 3x^2 + c`, where `c` is a real number, has three distinct `x`-intercepts

The set of all possible values of `c` is

A. `R`

B. `R^+`

C. `{0, 4}`

D. `(0, 4)`

E. `text{(−∞, 4)}`

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`D`

Show Worked Solution

`text(Consider the graph of)\ \ f(x) = x^3 – 3x^2`

`text(For three)\ xtext(-intercepts, translate up)`

`text{between (0, 4) units}`

`c ∈ (0,4)`

`=> D`

Algebra, MET2 2015 VCAA 6 MC

For the polynomial `P(x) = x^3 - ax^2 - 4x + 4,\ \ P(3) = 10,` the value of `a` is

A. `− 3`

B. `− 1`

C. `1`

D. `3`

E. `10`

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`C`

Show Worked Solution

`P(3)`	`= 3^3 – 3^2*a -4(3)+4`
`10`	`=27-9a-12+4`
`9a`	`=9`
`:. a`	`= 1`

`=> C`

Calculus, MET2 2013 VCAA 21 MC

The cubic function `f: R -> R, f(x) = ax^3-bx^2 + cx`, where `a, b` and `c` are positive constants, has no stationary points when

`c > b^2/(4a)`
`c < b^2/(4a)`
`c < 4b^2a`
`c > b^2/(3a)`
`c < b^2/(3a)`

Show Answers Only

`D`

Show Worked Solution

`text(If no stationary points,)`

♦♦ Mean mark 29%.

`=>\ text(No solution to)\ \ f{′}(x) = 0`

`f^{′}(x) = 3ax^2 -2bx +c`

`text(No solution when,)`

`Delta`	`< 0`
`(−2b)^2-4(3ac)`	`< 0`
`3ac`	`> b^2`
`:. c`	`> (b^2)/(3a)`

`=> D`