Question

f(x)=ln(4+2x-x^2); [-1,3] using rolle's theorem

Answer 1

By the rolles therom

if a real valued function f is continuous on a closed interval [a,b],differentiable on the open interval (a,b)

and f(a) = f(b),then there exist c in the open interval (a,b) such that f'(c) = 0

f(x) = ln(4+2x-x^2)

f(-1) = ln[4+2(-1)-(-1)^2]

= ln(4-2-1)

= ln(1)

= 0

f(3) = ln[4+2(3)-3^2]

= ln(4+6-9)

= ln(1)

= 0

here must exists c in the interval (-1 , 3) such that f'(c) = 0