00:01
This problem says to determine the location of each local extremum of the function, and we're given the function f of x equals x cubed plus 5 .5 x squared plus 6 x plus 4, and to figure out the possible locations of extremum we are going to find the first derivative of our f of x function, and we'll do that by using our power rules, which will bring our 3 to multiply by our understood coefficient of 1 to give us 3x, and then we'll take 1 away from our exponent to make that 3x squared.
00:25
We'll do the same thing with 2 and multiply by 5 .5 to give us 11 times x to the 2 minus 1 or x to the first.
00:32
The derivative of any linear term is just the coefficient, so that's 6, and the derivative of any constant is just 0, so the 4 is gone, and to find these possible location of extremum we will now set this equal to 0 and try to solve, and here we can't factor out the 3, so to be able to try to solve by factoring we'll have to multiply a times c, which would give us 18, and then we ask ourselves what two numbers would multiply to be 18 and add to be 11, and those two values are positive 9 and positive 2, but since we started off our factoring by multiplying a times c, we now have to divide by the a value into our numbers, and when we divide evenly we just change it to the whole number, so 9 thirds would simplify to plus 3.
01:15
If it won't simplify to a whole number, then we take our denominator and write it as the coefficient of our x value, so that would be 3x plus 2, with this still equal to 0.
01:25
So the two solutions we would get when we set x plus 3 equal to 0 is negative 3, and when we set 3x plus 2 equal to 0 we would get negative 2, and then divided by 3, or negative 2 thirds...