Let $f(x) = 2x - \tan x$ . Which pair of values shows that the smallest positive root of $2x - \tan x = 0$ lies between $x = 1$ and $x = 1.5$ ?

f(1) = -0.443

and

f(1.5) = 11.101

f(1) = 0.443

and

f(1.5) = 1.899

f(1) = 0.443

and

f(1.5) = -11.101

f(1) = 1.557

and

f(1.5) = 14.101

Explanation:

f(1)=2-\tan1=2-1.5574=0.443>0

and

f(1.5)=3-\tan1.5=3-14.101=-11.101<0

. The change of sign guarantees a root between 1 and 1.5.

Derived from ZIMSEC Mathematics Paper 1, November 2011, Q12

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