CAIE A-Level Mathematics 2.6.1 Root location

Practise proving where a root lies by sign changes or graph sketches, including single-root arguments from intersections.

Exam points

evaluate f(x) at two endpoints and use a sign change to justify a root interval
sketch suitable graphs on one diagram to show the number of roots in a given range

[Maximum number: 3]

Figure for Question 2.6.1—Root location question 1 — CAIE A-Level Mathematics AS

The diagram shows the curve with equation $y=\frac{2 \ln x}{3 x+1}$ . The curve crosses the x-axis at the point A and has a maximum point B. The shaded region is bounded by the curve and the lines x=3 and y=0.

Show by calculation that the x-coordinate of B lies between 3.0 and 3.1.

Equate numerator of first derivative to zero

M1

May be implied.

Consider sign of $\frac{2}{x}(3 x+1)-6 \ln x$ for 3.0 and 3.1

M1

OE

Obtain $0.074 \ldots$ and $-0.14 \ldots$ or equivalents and justify conclusion

A1

AG - necessary detail needed.
0.00075 and -0.001275 .

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2.6.1—Root location question 1