Use the Intermediate Value Theorem to determine whether the

Use the Intermediate Value Theorem to determine whether the polynomial function has a zero in the given interval.

1) f(x) = -7x4 + 4x2 + 7; [-2, -1]

2) f(x) = 8x3 - 9x - 10; [1, 2]

Answers

1) f(-2) = -89 and f(-1) = 4; yes

2) f(1) = -11 and f(2) = 36; yes

please show all work

Solution

Intermediate Value Theorem says that if f(x) is continuous on the interval [a, b] and k is a number that lies between f(a) and f(b), then there is at least one number c in [a, b] such that f(c) = k

1) f(x) = -7x⁴ + 4x² + 7; [-2, -1]

polynomial function f(x) = -7x⁴ + 4x² + 7 is continous over the interval [-2, -1]

f(-2) = -7(-2)⁴ + 4(-2)² + 7

f(-2) = -112 + 16+ 7

f(-2) = -89

f(-2)<0

f(-1) = -7(-1)⁴ + 4(-1)² + 7

f(-1) = -7 + 4+ 7

f(-1) = 4

f(-1)>0

f(x) changes its sign over the interval; [-2,-1]

-89<0<4

so by Intermediate Value Theorem polynomial function f(x) has a zero in the given interval.

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2)

f(x) = 8x³ - 9x - 10; [1, 2]

polynomial function f(x) =8x³ - 9x - 10 is continous over the interval [1,2]

f(1) =8*1³ - 9*1 - 10

f(1) = 8-9-10

f(1) = -11

f(1)<0

f(2) =8*2³ - 9*2 - 10

f(2) = 64-18-10

f(2) = 36

f(2)>0

f(x) changes its sign over the interval; [1,2]

-11<0<36

so by Intermediate Value Theorem polynomial function f(x) has a zero in the given interval.