A student answered the following question.
a and b are non-zero real numbers. Prove that the equation x3 + ax2 + b = 0 has three distinct real roots if 27b(b + 4a327) < 0.
Here is the student's solution:
I We differentiate y = x3 + ax2 + b to get dydx = 3x2 + 2ax = x(3x + 2a). Solving dydx = 0 shows that the stationary points are at (0, b) and (−2a3, b + 4a327).
II If 27b(b + 4a327) < 0, then b and b + 4a327 must have opposite signs, and so one of the stationary points is above the x-axis and one is below.
III If the cubic has three distinct real roots, then one of the stationary points is above the x-axis and one is below.
IV Hence if 27b(b + 4a327) < 0, then the equation has three distinct real roots.
Which one of the following options best describes the student's solution?
- AIt is a completely correct solution.
- BThe student has instead proved the converse of the statement in the question.
- CThe solution is wrong, because the student should have stated step II after step III.
- DThe solution is wrong, because the student should have shown the converse of the result in step II.
- EThe solution is wrong, because the student should have shown the converse of the result in step III.
Show the answer and worked solution
answer · E
- AIt is a completely correct solution.
- BThe student has instead proved the converse of the statement in the question.
- CThe solution is wrong, because the student should have stated step II after step III.
- DThe solution is wrong, because the student should have shown the converse of the result in step II.
- EThe solution is wrong, because the student should have shown the converse of the result in step III.
Steps I and II are fine, and end with "one stationary point above the axis and one below". To reach step IV the student needs that fact to imply three distinct real roots. But step III states the implication the other way round — three roots imply the stationary points straddle the axis. What is needed is the converse of step III.