Local extrema occur whenever the slope is equal to 0 so we must first find the derivative of the function, set it equal to 0, and then solve for x to find all x's for which there are local extrema.
Using the power-down rule we can find that f'(x)=8x^3-72x. Now set it equal to 0. 8x^3-72x=0. To solve, factor out an 8x to get 8x(x^2-9)=0 then using the rule of the difference of two squares split x^2-9 into its two factors to get 8x(x+3)(x-3)=0. Now set each of these separately equal to 0 because the entire expression will be 0 when any of the terms are 0.
This gives you 3 equations: 8x=0, x+3=0, and x-3=0. To solve the first one divide both sides by 8 to get x=0. For the second, subtract 3 from both sides to get x=-3. Lastly, for the third, add 3 to both sides to get x=3. These are all the x-values where local extrema will occur. Hope I helped!
