Solved on Feb 11, 2024

Continuous function f(x)f(x) has values {7,9,8,2,8}\{7, -9, 8, 2, -8\} at x={0,10,20,30,40}x=\{0, 10, 20, 30, 40\}. Guarantee roots between x=0x=0 and x=10x=10, x=20x=20 and x=30x=30, x=30x=30 and x=40x=40, but not between x=10x=10 and x=20x=20.

STEP 1

Assumptions
1. The function f(x)f(x) is continuous everywhere.
2. The table provides specific values of f(x)f(x) at x=0,10,20,30,40x = 0, 10, 20, 30, 40.
3. A root of the function f(x)f(x) is a value x=cx = c such that f(c)=0f(c) = 0.
4. The Intermediate Value Theorem states that if f(x)f(x) is continuous on the interval [a,b][a, b] and f(a)f(a) and f(b)f(b) have opposite signs, then there exists at least one cc in (a,b)(a, b) such that f(c)=0f(c) = 0.

STEP 2

Examine the values of f(x)f(x) at x=0x = 0 and x=10x = 10.
f(0)=7,f(10)=9f(0) = 7, \quad f(10) = -9

STEP 3

Since f(0)=7f(0) = 7 is positive and f(10)=9f(10) = -9 is negative, and f(x)f(x) is continuous everywhere, by the Intermediate Value Theorem, there must be at least one root between x=0x = 0 and x=10x = 10.

STEP 4

Examine the values of f(x)f(x) at x=10x = 10 and x=20x = 20.
f(10)=9,f(20)=8f(10) = -9, \quad f(20) = 8

STEP 5

Since f(10)=9f(10) = -9 is negative and f(20)=8f(20) = 8 is positive, and f(x)f(x) is continuous everywhere, by the Intermediate Value Theorem, there must be at least one root between x=10x = 10 and x=20x = 20.

STEP 6

Examine the values of f(x)f(x) at x=20x = 20 and x=30x = 30.
f(20)=8,f(30)=2f(20) = 8, \quad f(30) = 2

STEP 7

Since both f(20)=8f(20) = 8 and f(30)=2f(30) = 2 are positive, we cannot guarantee that there is a root between x=20x = 20 and x=30x = 30 using the Intermediate Value Theorem, as there is no sign change.

STEP 8

Examine the values of f(x)f(x) at x=30x = 30 and x=40x = 40.
f(30)=2,f(40)=8f(30) = 2, \quad f(40) = -8

STEP 9

Since f(30)=2f(30) = 2 is positive and f(40)=8f(40) = -8 is negative, and f(x)f(x) is continuous everywhere, by the Intermediate Value Theorem, there must be at least one root between x=30x = 30 and x=40x = 40.

STEP 10

Summarize the guarantees based on the Intermediate Value Theorem:
- Between x=0x=0 and x=10x=10, we guarantee there is a root \checkmark - Between x=10x=10 and x=20x=20, we guarantee there is a root \checkmark - Between x=20x=20 and x=30x=30, we cannot guarantee there is a root since there is no sign change - Between x=30x=30 and x=40x=40, we guarantee there is a root \checkmark

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