Consider the following statements Statement I If a₀+ a₁ 2 + a₂ 3 + .+ a_n n+1 =0 , where a₀, a₁, ., a_n are real numbers, then the polynomial a₀+a₁ x+a₂ x^2+ .+a_n x^n has a zero in the interval (0,1) . Statement II If f:[a, b] R is continuous on [a, b] and f is differentiable in (a, b) , where a>0 and if f(a) a = f(b) b , then there exists c (a, b) , such that c f^ (c)=f(c) . Which one of the following options is tr

Step-by-step solution

Let a polynomial function f(x)=a₀ x+a₁ x^2 2 +a₂ x^3 3 + +a_n x^ n+1 n+1 Now, as f(x) is a polynomial so it is continuous and differentiable in R . and as f(0)=0 and f(1)=a₀+ a₁ 2 + a₃ 3 + + 9 n n+1 =0 (given) So, according to Rolle's theorem f^ (x)=0 for at least one value of x (0,1) . Therefore, Statement (I) is true. It is given that [f: a, b] R is continuous on [a, b] and differentiable in (a, b) , where a>0 and f(a) a = f(b) b f(b)= b a f(a) Now, according to By Lagrange's mean value theorem (L.M.V.T), we have