Equations & Inequalities

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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