How do use the first derivative test to determine the local extrema #(x^2-10x)^4#?

Question

Evelyn Agard · Accepted Answer

The function has local (and global) minima at #x=0# and #x=10# and a local maximum at #x=5#.

Let #f(x)=(x^2-10x)^4#, then the Power Rule and Chain Rule give a derivative of #f'(x)=4(x^2-10x)^3*(2x-10)# #=8x^3(x-10)^3(x-5)# #f'(x)# is therefore clearly zero at #x=0,5,10#, and these are the critical points of #f#. A bit less clearly, #f'# changes sign from negative to positive as #x# increases through #x=0#, from positive to negative as #x# increases through #x=5#, and from negative to positive as #x# increases through #x=10#. You can plug numbers less than 0, between 0 and 5, between 5 and 10, and greater than 10 and use the continuity of #f'# to check this. You can also graph #f'# to confirm it graphically. The First Derivative Test then implies that the function has local (and global) minima at #x=0# and #x=10# and a local maximum at #x=5#. Interestingly, application of the Second Derivative Test in this example only allows us to conclude where the local maximum is, not where the local minima are (it turns out that #f''(0)=f''(10)=0# and #f''(5)=-125000<0#). In other words, the First Derivative Test is "stronger" (it can be used in more situations) than the Second Derivative Test.

Elijah Abram · Answer

undefined To use the first derivative test to determine the local extrema of $ f(x) = (x^2 - 10x)^4 $, follow these steps:

1. Find the first derivative of $ f(x) $, denoted as $ f'(x) $.
2. Set $ f'(x) $ equal to zero and solve for $ x $. These values of $ x $ are potential critical points.
3. Determine the sign of $ f'(x) $ in intervals separated by the critical points.
4. If $ f'(x) $ changes sign from positive to negative at a critical point, it indicates a local maximum at that point. If it changes sign from negative to positive, it indicates a local minimum.
5. If $ f'(x) $ does not change sign at a critical point, the test is inconclusive for that point.

To execute these steps specifically for the function $ f(x) = (x^2 - 10x)^4 $:

1. Calculate $ f'(x) $.
2. Solve $ f'(x) = 0 $ for $ x $ to find critical points.
3. Examine the sign of $ f'(x) $ in intervals defined by the critical points.
4. Determine the behavior of $ f(x) $ at each critical point according to the first derivative test.