Question

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

f(x)=x^5-x^4+9x^3-5x^2-16x+5;[1.3,1.6]

f(x)1.3= ? simplify answer
f(x)1.6= ? "

Answer 1

1.3^5 = 3.71293----> 3.71
1.3^4 = 2.8561 ---->-2.86
1.3^3 = 2.197 ----->19.77
1.3^2 = 1.69------->-8.45
-16*1.3 ----------->-20.8
+5 ----------------> +5
add
f(1.3) = -3.63
now do f(1.6)
1.6^5 = 10.48576 ----> 10.5
1.6^4 = 6.5536 ------>-6.55
1.6^3 = 4.096 -------> 36.9
1.6^2 = 2.56 -------->-12.8
1.6*-16 ------------->-25.6
+5------------------->+5
add
f(1.6) = 7.45
ah ha!
somewhere in between there this function which is continuous crossed the x axis at least once to get from -3.63 to + 7.45
Therefore it had at least one zero in there.