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Calculus, 2ADV C3 2018 HSC 14c

Let  `f(x) = x^3 + kx^2 + 3x - 5`, where `k` is a constant.

Find the values of `k` for which  `f(x)`  has NO stationary points.  (3 marks)

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`-3 < k < 3`

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`f(x) = x^3 + kx^2 + 3x – 5`

♦ Mean mark 49%.

`f prime(x) = 3x^2 + 2kx + 3`
 

`text(No S.P.’s exist if)\ \ f prime(x)\ \ text(has no roots,)`

`Delta` `< 0`
`b^2 – 4ac` `< 0`
`(2k)^2 – 4 xx 3 xx 3` `< 0`
`4k^2 – 36` `< 0`
`k^2 – 9` `< 0`
`(k – 3) (k + 3)` `< 0`

 

`:. -3 < k < 3`

Calculus, 2ADV C3 2014 HSC 14a

Find the coordinates of the stationary point on the graph  `y = e^x − ex`, and determine its nature.   (3 marks)

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`(1,0)\ =>text(MINIMUM)`

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`y` `= e^x – ex`
`dy/dx` `= e^x – e`
`(d^2 y)/(dx^2)` `= e^x`

 
`text(S.P. when)\ \ dy/dx = 0`

`e^x – e` `= 0`
`e^x` `= e^1`
`x` `= 1`

 
`text(At)\ \ x = 1`

`y` `= e^1 – e = 0`
`(d^2 y)/(dx^2)` `= e > 0\ \  => text(MIN)`

 
`:.\ text(MINIMUM S.P. at)\ (1,0)`

Calculus, 2ADV C3 2010 HSC 8d

Let  `f(x) = x^3-3x^2 + kx + 8`, where `k` is a constant.

Find the values of `k` for which `f(x)` is an increasing function.   (2 marks)

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`k>3`

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`f(x)` `= x^3-3x^2 + kx + 8`
`f^{′}(x)` `= 3x^2-6x + k`

  
`f(x)\ text(is increasing when)\ \ f^{′}(x) > 0`

`=> 3x^2-6x + k > 0`

♦♦ Mean mark 28%.
MARKER’S COMMENT: The arithmetic required to solve `36-12k<0`  proved the undoing of many students.

 

`f^{′}(x)\ text(is always positive)`

`=> f^{′}(x)\ text(is a positive definite.)`

`text(i.e. when)\ \ a > 0\ text(and)\ Delta < 0`
 

`a=3>0`

`Delta = b^2-4ac`

`:. (-6)^2-(4 xx 3 xx k)` `<0`
`36-12k` `<0`
`12k` `>36`
`k` `>3`

 

`:.\ f(x)\ text(is increasing when)\ \ k > 3.`

Calculus, 2ADV C3 2013 HSC 12a

The cubic  `y = ax^3 + bx^2 + cx + d`  has a point of inflection at  `x = p`. 

Show that  `p= - b/(3a)`.   (2 marks)

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 `text(Proof)\ \ text{(See Worked Solutions)}`

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`text(Show)\ \ p= – b/(3a)`

`y` `=ax^3 + bx^2 + cx + d`
`y prime` `=3ax^2 + 2bx + c`
`y″` `=6ax + 2b`

 
`text(Given P.I. occurs when)\ \ x = p`

`=> y″=0\ \ text(when)\ \ x=p`

`:.\ 6ap + 2b` `=0`
`6ap` `=-2b`
`p` `= -(2b)/(6a)`
  `=-b/(3a)\ \ \ text(… as required)`