Equations & Inequalities

When you are told to solve an equation you usually hunt for x by calculation, but derivatives let you turn the equation into a single picture. The real roots of x³−3x=k are exactly the points where the curve y=x³−3x meets the horizontal line y=k, so you can count the roots by eye instead of solving. Inequalities work the same way: you only need to check that the minimum of the difference between the two sides stays at or above 0. Slide the horizontal line y=k up and down to see why the boundary where the count jumps from one root to three sits right at the local maximum and minimum.

Real Roots Are Graph Intersections

Horizontal line y = k0

👀 See It

①The real roots of f(x)=k are the intersections of y=f(x) and y=k

②Raising or lowering k switches the count between 1 and 3

③The boundaries are the moments it passes the local max and min

Counting Roots of a Cubic

Judge by the sign of extrema

(local max) × (local min) < 0 ⇒ three distinct real roots

If the product is 0, a double root (2); if greater than 0, just 1

Separate Into the Form f(x)=k

Constant separation

number of roots of f(x)=k = intersections of y=f(x) with y=k

Seen as a moving horizontal line y=k, the extrema become the boundaries

Solve It Directly

Example 1

Find the number of distinct real roots of x³−3x+1=0.

f(x)=x³−3x+1, and f'(x)=3x²−3=0 gives x=±1.

Local max f(−1)=3>0, local min f(1)=−1<0 — opposite signs.

(local max)(local min) < 0 ⇒ 3 real roots

▸ 3

For a cubic, the signs of the local max and min give the root count immediately.

Example 2

Show that x³−3x+2 ≥ 0 for all x ≥ 0.

g(x)=x³−3x+2, and g'(x)=3x²−3=0 gives x=1 within x≥0.

On x≥0 the minimum is g(1)=1−3+2=0 ≥ 0.

min_x≥0 g(x) = g(1) = 0 ⇒ g(x) ≥ 0

▸ minimum 0 ≥ 0, so it holds

Proving an inequality means showing the minimum of (LHS−RHS) is at least 0.

Wrap-up

Core strategy

root count = graph intersections, inequality = sign of the minimum

Equations via the sign of extrema, inequalities via the minimum — both solved by differentiation

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

f(x)=x³−3x, and f'(x)=3x²−3=0 gives local max f(−1)=2, local min f(1)=−2.

Three roots require the line y=k to lie between the local min and max.

−2 < k < 2

🎯 Exam Points

①Reduce equation roots to graph intersections

②A cubic’s root count comes from the sign of (max)(min)

③For f(x)=k, separate the constant and move a horizontal line

④For inequalities, show the minimum of the difference ≥ 0

⑤With a restricted domain, use the minimum on that interval

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