Asked to solve an equation, people usually hunt for x by calculation, but differentiation lets you turn the equation into a single graph.
The real roots of x³ − 3x = k are the x-coordinates where the curve y = x³ − 3x meets the horizontal line y = k, so the number of roots can be counted without calculation.
Inequalities work the same way: it is enough to check that the minimum of the difference between the two sides is at least 0.
In the picture, moving the line y = k up and down shows that the switch from three meeting points to one happens at the local maximum 2 and the local minimum −2.
Real roots are meeting points of graphs
seegongsik.com
The points (gold) where y = x³ − 3x meets the horizontal line y = k; their number is the number of distinct real roots of x³ − 3x = k
0
👀 What to read in the picture
①The real roots of f(x) = k are the x-coordinates where y = f(x) meets y = k
②For −2 < k < 2 there are 3 meeting points; for k > 2 or k < −2 there is 1
③At k = ±2 the line touches a peak or a valley, giving a double root, so there are 2 distinct real roots
④The values of k at the switch are the local maximum and minimum
How many real roots a cubic has
Decide by the signs of the extrema
(local max) × (local min) < 0 ⇒ three distinct real roots
If the product is 0 there is a double root and 2 distinct real roots; if it is positive, 1
Rewriting as f(x) = k
Separating the constant
number of roots of f(x)=k = intersections of y=f(x) with y=k
Seen as a horizontal line moving up and down, the extrema become the points where the count changes
Worked examples
Example 1
Find the number of distinct real roots of x³−3x+1=0.
1
Let f(x) = x³ − 3x + 1; then f'(x) = 3x² − 3 = 0 gives x = ±1.
2
The local maximum f(−1) = 3 > 0 and the local minimum f(1) = −1 < 0 have opposite signs.
(local max)(local min) < 0 ⇒ 3 real roots
▸ 3
The peak is above the x-axis and the valley below it, so the graph crosses the x-axis three times. It is the same situation as k = −1 in the picture.
Example 2
For x ≥ 0, show that x³ − 3x + 2 ≥ 0.
1
Let g(x) = x³ − 3x + 2; from g'(x) = 3x² − 3 = 0, the root with x ≥ 0 is x = 1.
2
For x ≥ 0 the minimum of g is g(1) = 1 − 3 + 2 = 0, which is at least 0.
minx≥0 g(x) = g(1) = 0 ⇒ g(x) ≥ 0
▸ minimum 0 ≥ 0, so it holds
An inequality is proved by showing that the minimum of (left side − right side) is at least 0. Factoring g(x) = (x − 1)²(x + 2) also shows it is non-negative for x ≥ 0.
Summary
Strategy
root count = graph intersections, inequality = sign of the minimum
Equations by the signs of the extrema, inequalities by the minimum
2022 provincial mock exam Math type, adapted
For which range of k does x³−3x = k have three distinct real roots?
①k < −2
②−2 < k < 2
③k > 2
④k = ±2
⑤all real numbers
▸ ② −2 < k < 2
1
For f(x) = x³ − 3x, f'(x) = 3x² − 3 = 0 gives the local maximum f(−1) = 2 and the local minimum f(1) = −2.
2
Three distinct real roots need the line y = k to lie between the local minimum and the local maximum.
−2 < k < 2
🎯 Exam Points
①Count real roots as meeting points of graphs
②The number of real roots of a cubic follows from the sign of (local max)(local min)
③Separate the constant as f(x) = k and move a horizontal line
④For an inequality, show that the minimum of the difference is at least 0
⑤If x is restricted, look at the minimum within that range