List of critical numbers 31 1 Find all critical values for t

List of critical numbers:

3-1 #1

Find all critical values for the function and then list them (separated by commas) in the box below. List of critical numbers:

Solution

First we find the critical points; points where f \' (x) = 0 or Does not exist

f\'(x) = (2(x²+x+1) - (2x +1)(2x+8)) / (x²+x+1)²

For f\'(x) = 0
We have 2x² + 2x + 2 - 4x² -18x - 8 = 0
Hence : 2x² +16x + 6 = 0
or x² + 8x + 3=0
Or x = -0.394, -7.606

Now the critical values = f(critical points) = (2*(-0.394) + 8)/(0.394² -0.394 + 1) = 9.474
and (2*(-7.606) + 8)/(7.606² -7.606 + 1) = -6.731

Hence the critical values are  9.474,-6.731