 Hello friends welcome to the session I am Malka and going to help you solve the question that is if f is a function the closed interval minus 5, 5 to r is a different symbol function and if f dash x does not vanish anywhere then proof it f minus 5 is not equal to f5. So let's start with the solution let function y equal to fx. Now since we are given that the function is a differentiable function therefore we can say that it is a differentiable function therefore it is continuous closed interval minus 5, 5. Now this implies that x means the element of closed interval minus 5, 5 such that f dash x equal to fb minus fa upon v minus a this implies that f dash x is equal to f5. Now since we are given in the question that f dash x does not vanish that is f dash x is not equal to 0 therefore we can write that f5 minus f minus 5 upon 10 is not equal to 0. So we can say that f5 minus f minus 5 is not equal to 0 this implies that minus 5 is not equal to f5 so which is the required solution hope you understood it and enjoy the session goodbye and take care.