Question 1079041: Please help me solve this
Consider the function f(x) = x3 + px + q.
(a) Determine the values of p and q if f(x) has a stationary point at (−2,3).
(b) Show, using calculus, that there is a second stationary point at (2,−29), and classify both
stationary points.
(c) Determine f′′(x) and hence show that there is a non-stationary point of inflection and determine its coordinates.
(d) For what values of k would the equation f(x) = k have 3 distinct solutions. Give reasons for your answer.
(Hint: Sketch the graph of y = f(x)
Answer by Boreal(15235)   (Show Source):
You can put this solution on YOUR website! Derivative of the function is 3x^2+p
Set that equal to 0 and x^2=-p/3 at the stationary point.
F(-2)=-8-2p+q=3
q-2p=11
x^2=-p/3, 4=-p/3 and p=-12;
Therefore q=-13
x^3-12x-13
3x^2-12=0
x=+/-2
when x=2, f(2)=8-24-13=-29, so (2,-29) is a point.
The second derivative is 6x.
When x<0, the second derivative is negative and there is a maximum.
When x>0, it is positive and there is a minimum.
the inflection point occurs at (0, -13), for there the second derivative changes sign.
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