Number of real solutions of an equation

When a cubic has a local maximum and a local minimum, move the horizontal line $y=k$ up and down and count how many times it meets the graph.

If $k$ equals the local maximum or minimum, the line is tangent to the graph and there are 2; if $k$ is above the local maximum or below the local minimum, there is 1. For example, for a cubic with a local maximum $5$ at $x=2$ and a local minimum $1$ at $x=4$, the right side $0$ of $f(x)=0$ is below the local minimum $1$, so there is 1 real solution.

Examples

1. 01A cubic with a local maximum 1 at x=0 and a local minimum -2 at x=2. What goes in □ so that there are 3 solutions?$\displaystyle \square < k < 1$AnswerHide

   Answer$\displaystyle -2$

   Steps

   1. Three solutions means k lies between them
   2. $\displaystyle -2$$\displaystyle < k < 1$
   3. The bound is the minimum value, not its x = 2
2. 02A cubic with a local maximum 7 at x=1 and a local minimum 3 at x=4. For which k are there 3 solutions?$\displaystyle f(x)=k$AnswerHide

   Answer$\displaystyle 3<k<7$

   Steps

   1. Check what happens at each end
   2. $\displaystyle k = 7$$\displaystyle \Rightarrow 2$$\displaystyle k = 3$$\displaystyle \Rightarrow 2$
   3. At the ends it touches and gives 2, so the inequality is strict
3. 03A cubic with a local maximum 5 at x=2 and a local minimum 1 at x=4. How many real solutions?$\displaystyle f(x)=0$AnswerHide

   Answer$\displaystyle 1$
4. 04A cubic with a local maximum 3 at x=-1 and a local minimum 1 at x=1. What goes in □ so that there are 3 solutions?$\displaystyle \square < k < 3$AnswerHide

   Answer$\displaystyle 1$

   Steps

   1. Three solutions means k lies between them
   2. $\displaystyle 1$$\displaystyle < k < 3$
   3. The bound is the minimum value, not its x = 1
5. 05A cubic with a local maximum 9 at x=-1 and a local minimum 1 at x=1. What goes in □ so that there are 3 solutions?$\displaystyle \square < k < 9$AnswerHide

   Answer$\displaystyle 1$

   Steps

   1. Three solutions means k lies between them
   2. $\displaystyle 1$$\displaystyle < k < 9$
   3. The bound is the minimum value, not its x = -1
6. 06A cubic with a local maximum -1 at x=0 and a local minimum -4 at x=3. What goes in □ so that there are 3 solutions?$\displaystyle \square < k < -1$AnswerHide

   Answer$\displaystyle -4$

   Steps

   1. Three solutions means k lies between them
   2. $\displaystyle -4$$\displaystyle < k < -1$
   3. The bound is the minimum value, not its x = 3

FAQ

Q1Why rewrite the equation as $f(x)=k$?AnswerHide

A

With $k$ alone on the right, the graph on the left is fixed no matter what $k$ is, and $k$ is just the height of a line. Draw the graph once, then slide the line up and down to read the number of solutions for each $k$.

Q2How many solutions are there when $k$ equals the local maximum?AnswerHide

A

Two distinct real solutions. The line $y=k$ touches the graph at the local maximum (a double root) and crosses it at one other point.