Solving Calculus Problems with Intermediate Value and Rolle's Theorems

[oahsen]

good days. i have two problems about calculus. could u help me ;
first is that;
consider the equation cot(pi*x)=x;
show t
hat this equation has exactly one solution rk in the Ik=(k,k+1) for every k<Z ( k<Z = k is a integer)

second is that ; show that for any differentiable function f(x) in the real line such that f(0)=0 , f(1)=1 , f(2)=4, there exist a point c in (0,2) such that f''(c)=2.

i think both of the problems are about the inter.value theo,mean value theo. and rolle theo. but i haven't any idea of how can i solve.thanks for help...

 

[lippyka]

For the first problem, you can use the Intermediate Value Theorem. Since cot(pi*x) is continuous, it takes on all values between x and -x in the interval (k, k+1). Thus, there must be a point rk in (k, k+1) such that cot(pi*rk) = rk. For the second problem, you can use Rolle's Theorem. Since f(x) is differentiable and satisfies the given conditions, then by Rolle's Theorem, there must be at least one c in (0,2) such that f'(c) = 0. Since f(x) is also twice differentiable, then by Rolle's Theorem again, there must be at least one c in (0,2) such that f''(c) = 2.