Page 14 - Pra U STPM 2022 Penggal 2 - Mathematics

Mathematics Semester 2 STPM Chapter 1 Limits and Continuity

Example 9

1 Use the intermediate value theorem to show that the equation x – x + 2 = 0 has at least one real root.
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Solution: Let f(x) = x – x + 2 f(x)
f(x) is continuous since it is a polynomial
f(–1) = 2 2
f(–2) = –28
c
Since f(–2) , 0 and f(–1) . 0, the intermediate –2 –1 0 x
value theorem tells us that f(c) = 0 for some c
in the interval (–2, –1).
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Hence, the equation x – x + 2 = 0 has at least
one real root.
Graph of f in the interval [–2, –1].

–28

Example 10

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Given f(x) = x – x + x. Show that there is a number c  R such that f(c) = 10.
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Solution: f(0) = 0 and f(3) = 21.
We have f(0) , 10 , f(3)
Since f is continuous, there must be a number c  R such that f(c) = 10.

Example 11

Show that there is a positive real root of the equation x = x + 1.
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Solution: Let f(x) = x – x – 1
f(0) = –1
f(2) = 125

Since f(0) , 0 and f(2) . 0, the intermediate value theorem tells us that f(c) = 0
for some c in the interval (0, 2), all of whose elements are positive numbers.
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Hence, there is a positive real root of the equation x = x + 1.

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01 STPM Math(T) T2.indd 12 28/01/2022 5:30 PM