HardMathematicsHigh School (11–12)Functions and Calculus › Continuity and DifferentiabilityNumericalBoard Exam

Let g(x) = x³ - 12x + 8. A cubic equation has at most three real roots…

Try it on paper first. A hint, the key idea and the worked solution are here when you need them.

Let $g(x) = x^3 - 12x + 8$ . A cubic equation has at most three real roots. Find $g(0)$ , $g(1)$ , $g(3)$ and $g(4)$ . Use the signs, together with $g(-4) = -8$ and $g(-3) = 17$ , to show that $g(x) = 0$ has exactly three real roots. Give the number of real roots.

Need a reminder? The key ideaShow

If $f$ is continuous on $[a, b]$ and $f(a)$ and $f(b)$ have opposite signs, then $f(c) = 0$ for some $c$ in $(a, b)$ . Halving the interval and keeping the half with the sign change traps the root.

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.

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