Answer to Question #158000 in Calculus for Cypress

Question

Deduce that f(x)=x^(3)+3x^(2)-1 has exactly one zero in each of the intervals [0,1],[-1,0] and [-3,-2]

Answer 1

to check that function has zero on the given intervals, we have to check if it has different signs on them

f(0) = -1 f(1) = 3

f(-1) = 1 f(0) = -1

f(-3) = -1 f(-2) = 3

as we can see 3 intervals satisfies given condition

it means that our function has at least 1 zero at each of the interval, than at least 3 zeros on these 3 intervals

and as a function f(x) has power of 3 it means that it has exactly 3 zeros and 3 of them are on the intervals (one zero for each interval)