Cubic with a Parameter: Exactly Two Real Roots
A curve C has equation y = x^3 - 6x^2 + 9x + k, where k is a real constant. A student is asked to determine the values of k for which C has exactly two distinct intersections with the x-axis. They differentiate and find stationary points at x = 1 and x = 3. For k = 0 they sketch the curve and locate a local maximum at (1, 4) and a local minimum at (3, 0). The student concludes that exactly two distinct roots occur only when the local minimum touches the x-axis, so k = 0. A classmate says the local maximum case also needs checking and that the answer should be a set of values, not a single number. The student must reconcile the sketch with the algebra, without a calculator, and give exact values.
1. Find the coordinates of the stationary points of C in terms of k, and classify each stationary point.
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Differentiate: f'(x)=3x^2-12x+9=3(x-1)(x-3). So stationary points occur at x=1 and x=3.
Substitute: f(1)=1-6+9+k=4+k, and f(3)=27-54+27+k=k. Thus the stationary points are (1, 4+k) and (3, k).
Classify using the second derivative: f''(x)=6x-12. At x=1, f''(1)=-6<0, so (1, 4+k) is a local maximum. At x=3, f''(3)=6>0, so (3, k) is a local minimum.
2. Determine all values of k for which C has exactly two distinct real roots. Justify both repeated-root cases and check that each really gives two distinct roots.
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A cubic has exactly two distinct real roots only if one root is repeated and the third root is different, so one stationary value must be zero.
If the local maximum is zero, 4+k=0, so k=-4. Then f(x)=x^3-6x^2+9x-4=(x-1)^2(x-4). The distinct roots are x=1 and x=4, and the local minimum is f(3)=-4<0, so the curve crosses again. This is valid.
If the local minimum is zero, k=0. Then f(x)=x(x-3)^2. The distinct roots are x=0 and x=3, and the local maximum is f(1)=4>0. This is also valid.
Therefore k=-4 or k=0. Both repeated-root cases have been checked to give exactly two distinct roots, not one.
3. Find the set of k for which C has three distinct real roots, and the set for which C has exactly one real root.
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Three distinct real roots require the local maximum above the x-axis and the local minimum below it: 4+k>0 and k<0. Hence -4<k<0.
One real root occurs when both stationary values are on the same side of the x-axis: k>0 makes both 4+k and k positive, while k<-4 makes both negative. At k=-4 and k=0 there are exactly two distinct roots, so these endpoints are excluded.
Thus: three distinct real roots for -4<k<0; one real root for k<-4 or k>0.