 In this video, we provide the solution to question number 16 for the practice final exam for math 1210 We're given the function f of x equals x to the fourth cube plus 36x squared plus 11 We're given the domain x ranges from negative one to one close interval and we're asked to find the absolute maximum and absolute minimums of this function So this is an extreme value problem We're gonna first look for the critical number. So we have to compute the derivative of f of x by the usual derivative rules We're gonna get f prime of x equals 12x squared plus 72x the derivative of 11 of course is 0 and we need to set this equal to 0 to find the critical numbers There's no place where this derivative would be undefined Factoring out the least are the grace common divisor. Excuse me. There's a coefficient of 12x Well, there's a coefficient amongst the co excuse me. There's common factor amongst the coefficients of 12 We can also factor on x. So the gcd is gonna be 12x that leads behind an x plus 6 is equal to 0 So our critical numbers turn out to be x equals 0 which comes from the 12x if you said that equal to 0 And then you're gonna get a negative 6 if you from the factor x plus 6 now x equal negative 6 we can discard that one because that's not inside of our domain We are critical numbers have to be between negative 1 and 1 so we're gonna build a t-chart using the boundary points negative 1 and 1 we're also gonna use the critical number 0 which lives inside of that Interval and we have to evaluate the function not the derivative at those values So if we plug in negative 1 into the function, we're gonna get f of negative 1 which equals negative 4 Plus 36 plus 11 that's gonna turn out to be 43 We have to compute f of 0 which is gonna be 11 and then we have to compute f of 1 Which is gonna be 4 plus 36 plus 11 that turns out to be 51 Now the smallest number amongst this list is going to be the absolute minimum and we see that is gonna be y equals 11 So the absolute Minimum of this function will be y equals 11 which is obtained at x equals 0 We also see that the largest number amongst this list will be the absolute maximum so that's gonna be 51 So the absolute maximum is obtained excuse me the absolute maximum is y equals 51 And it's obtained at the number x equals 1