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A designer claims that x³ + 3x + 5 = 0 has exactly one real solution…
Contributed by StemPy LabTry it on paper first. A hint, the key idea and the worked solution are here when you need them.
A designer claims that has exactly one real solution. Let . Find how many real roots has. Then find and , and give the number of real roots of .
Need a reminder? The key ideaShow
Rolle's theorem: if is continuous on , differentiable on and , then for some in . Check all three conditions, then solve and keep the solutions inside .
The worked solution
Have you tried it?
Work it out on paper first. It is fine to get stuck or make a mistake: that is how we learn. When you are ready, go through the solution one step at a time, and check each step against your own.
Step 1. The derivative. , so has real roots.
Step 2. At most one root. If had two roots, Rolle's theorem would give a point between them with . That is impossible, so has at most one real root.
Step 3. At least one root. and . The signs differ and is continuous, so a root lies in .
Step 4. Together. Exactly one real root.
Answer: , , and .
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