Pearson Edexcel IAL Mathematics P3.6.1 Locating roots using sign changes

Practise proving that roots lie in stated intervals by evaluating f(x), detecting sign changes and justifying continuity.

Exam points

evaluate f(a) and f(b) to show opposite signs across a given interval
state that f(x) is continuous before concluding that a root lies in the interval
choose a narrow interval to justify a root value correct to 3 or 4 decimal places

[Maximum number: 2]

Figure 1

Figure 1 shows a sketch of part of the curve with equation y=f(x) where

\mathrm{f}(x)=\frac{2 x^{2}+3 x-4}{\mathrm{e}^{x}}-\frac{1}{x^{2}} \quad x \in \mathbb{R} \quad x \neq 0

Show that f(x)=0 has a root $\alpha$ in the interval [1,2]

$f(1)=\frac{2(1)^{2}+3(1)-4}{e^{1}}-\frac{1}{1^{2}}=\ldots \text { and } f(2)=\frac{2(2)^{2}+3(2)-4}{e^{2}}-\frac{1}{(2)^{2}}=\ldots$ $\mathrm{f}(1)=-0.6(321 \ldots)[<0]$ and $\mathrm{f}(2)=1.(103 \ldots)[>0]$
⇒ There is a sign change and f(x) is continuous over the
interval hence root (in the interval [1,2] )

M1

(2)

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P3.6.1 - Locating roots using sign changes question 1