 and welcome to the session, let us discuss the following question, question says, verify a rose theorem for function fx is equal to x square plus 2x minus 8, x belongs to closed interval minus 4, 2. First of all, let us understand that if we have given a function f from closed interval a b to it is continuous on closed interval a b and differentiable on open interval a b such that f a is equal to f b, then there exists c belonging to open interval a b such that f dash c is equal to 0, this is known as rose theorem, we will use it as a key idea to solve the given question. Let us now start with the solution, we are given fx is equal to x square plus 2x minus 8 and x belongs to closed interval minus 4, 2 clearly we can see this is a polynomial function and polynomial function is continuous at every real number, so fx is equal to x square plus 2x minus 8 is continuous on closed interval minus 4, 2. Now, let us find out f dash x differentiating both sides of this expression with respect to x, we get f dash x is equal to 2x plus 2, now clearly we can see given function is differentiable on open interval minus 4, 2. So, we can write fx is equal to x square plus 2x minus 8 is differentiable on open interval minus 4, 2. Now, we will find out f minus 4, this is equal to minus 4 square plus 2 multiplied by minus 4 minus 8 which is equal to 16 minus 8 minus 8, simplifying we get f minus 4 is equal to 0. Now, let us find out value of f2, this is equal to 2 square plus 2 multiplied by 2 minus 8, this is equal to 4 plus 4 minus 8, simplifying we get f2 is equal to 0, so we get f minus 4 is equal to f2 is equal to 0. Now, according to Roll's theorem where exists c belonging to open interval minus 4, 2 such that f dash c is equal to 0, now since f dash x is equal to 2x plus 2, we get c is equal to minus 1, for c is equal to minus 1 f dash c is equal to 0, so we can write at c is equal to minus 1, f dash c is equal to 0 and clearly we can see minus 1 belongs to open interval minus 4, 2 the Roll's theorem is verified, this completes the session, hope you understood the session, take care and goodbye.