Find the values of x for which f(x) is an increasing function given that f(x)=8x-2x^2
When a function is increasing, it’s derivative is positive. So first let us differentiate f(x). To differentiate xn we multiply by n and then reduce the power by 1. So f’(x)=8-2*2x=8-4x. We want to find the values of x for which the derivative is positive i.e. f’(x)>0. So we want to find the values of x for which 8-4x>0. We need to rearrange the inequality to isolate x. Firstly, subtract 8 from both sides to give -4x>-8, then divide both sides by -4, making sure to reverse the inequality sign since we are dividing by a negative number, to get x<2. Hence our function has a positive derivative when x is less than 2, so our function f(x) is increasing for all x<2.
