Use the intermediate value theorem to show that the polynomial function has a zero in the given interval

Question

Find the value of f(1.2)

Find the value of f(1.5)

f(x)=x^5-x^4+8x^3-4x^2-18x+7;[1.2,1.5]

Answer 1

The polynomial function f(x) = x⁵ - x⁴ + 8x³  - 4x² - 18x + 7

f(1.2) = (1.2)⁵ - (1.2)⁴ + 8(1.2)³  - 4(1.2)² - 18(1.2) + 7

= 2.48832 - 2.0736 + 13.824 - 5.76 - 21.6 + 7

= - 6.12128

f(1.2) < 0

f(1.5) = (1.5)⁵ - (1.5)⁴ + 8(1.5)³  - 4(1.5)² - 18(1.5) + 7

= 7.59375 - 5.0625 + 27 - 9 - 27 + 7

= 0.53125

f(1.5) > 0

From intermediate mean value theorem

Since f(1.2) < 0 and f(1.5) > 0

The function has go through zero at some point in the interval [1.2, 1.5].