 In this video, we provide the solution to question number three for practice exam four for math 12 10 In which case we are given the function f of x equals five-thirds x cube plus 19 Havs x squared minus 4x plus 11 and we're asked on which of the following intervals is f Decreasing well if we want to figure out where the functions decreasing then we need to figure out where its derivative is Negative so we need to calculate the derivative of the function now these these fractions here might seem a horn to many of us But it actually turns out it's a saving grace for us if we calculate the derivative by the usual power rule and such The derivative is going to look like five-thirds x cube. Well the derivative x cubed is a 3x squared So we're going to get 5x squared when you take the derivative of 19 halves times x squared You're going to get 19x and the derivative of negative 4x becomes a negative 4 and the derivative of 11 since it's constant is going to turn out just to be a zero So we need to solve where the derivative is negative. We have to figure out where the drips equal to zero So we have to figure out what the derivatives x intercepts are that is what are the critical numbers So we set this thing equal to zero. Well, since we have a quadratic function It makes sense that we could use the quadratic formula We can complete the square or we could try to factor it if possible I'm going to try factoring it notice if I take the first and last coefficient multiply them together four times Excuse me five times negative four is equal to negative 20 I need factors of negative 20 that out to be 19. So my suggestion would be 20 and negative one Let's try to use those factors You could try to guess and check a little bit if you wanted to as well But I might try to factor this by groups right here. So we have 5x squared plus 20x That's gonna be my first group and then you're going to get negative x minus four as my second group So factoring these things by groups then the first group you can take out of 5x that leaves behind x plus 4 From the second one take away a negative one that leaves behind x plus 4 and so we see the derivative F prime of x is gonna look like 5x minus 1 times x plus 4 This is important Because with the derivative in hand we can then find the critical numbers which we get one fifth and negative 4 So we can now compare our sign chart in which case we're gonna label the sign chart based upon these critical numbers We get negative 4 and 1 fifth. So we have to identify. Where is the derivative? negative we can plug in test values like we could pick zero which is between negative 4 and 1 fifth we pick one That's bigger negative 10. You can plug in some test values There's many ways one could approach this the fact that the derivative is quadratic and the leading coefficient here is a 5 I know my parabola is going to look something like this It's concave up because the leading coefficient is positive And therefore it's the first derivative is going to look positive negative positive So where is the first derivative negative? That will be between negative 4 and 1 fifth That'll be exactly where the function's decreasing And so we see that the correct answer would then be b the interval from negative 4 to 1 fifth