Question

Use the Intermediate Value Theorem to check whether the equation x^3–3x+2.1=0 has a root in the interval (0,1)
answer: yes or no ?

i have no idea how to answer to go about solving this question, thanks for the help!

Answer 1

if x = 0 the function is 2.1
if x = 1 , the function is 0.1
so it does not necessarily cross the x axis between those points

If it does cross, it must cross twice to go from + to - to + again
so it would have to have a slope of zero in between there
df/dx = 3 x^2 -3
0 = x^2 - 1
x= +1 or -1 for 0 slope
so it has no zero slope points between 0 and 1
So
It never crosses the x axis between 0 and 1