Ross Cross

01/09/2023 · Middle School

(a) Let \( f \) be a continuous function on \( [0,1] \) and differentiable on \( (0,1) \). Let also \( f(0)= \) \( f(1)=0 \). Prove that there exists a point \( c \in(0,1) \) such that \( f^{\prime}(c)=f(c) \). (Hint use \( \left.g(x)=f(x) e^{-x}\right) \). (b) If \( f^{\prime}(x)=0 \) at each point in an open interval \( (a, b) \). Prove that \( f \) is a constant on the interval \( (a, b) \). (c) Show that the equation \( \cos x=x \) has a root between 0 and 1 . (cos,\( =0 \) (d) Where is \( f(x)=x e^{2 x} \) increasing or decreasing?