Consider the function f(x)x^4 -32x^2 +2, -3 < or = to x < or = to 9. This function has an absolute minimum value equal to ? and an absolute maximum value equal to ?
do the first derivative, set equal to zero.
Then at each zero, test the second derivative.
Post back if you are lost.
As in the previous problem (quadratic), you will need to first find f'(x).
Equate f'(x) to zero and solve for the roots. Even f'(x)=0 is a cubic equation, you can easily solve for the three roots by factorization.
Two of the three roots fall within the given interval, so there is one maximum and one minimum, which you can check by evaluating f"(x). F"(x)>0 is a minimum and f"(x)<0 is a maximum.
Evaluate f(x) at the two limits of the interval, as well as at the roots of f'(x)=0. These four values will give the absolute maximum and absolute minimum within the interval.
Post your answers for checking if you wish.
To find the absolute minimum and maximum values of the function f(x) = x^4 - 32x^2 + 2 over the given interval [-3, 9], we can follow these steps:
1. First, calculate the derivative of the function f(x) with respect to x. Let's call this derivative function f'(x).
f'(x) = 4x^3 - 64x
2. Now find the critical points of the function, which occur where the derivative is either zero or undefined. To do this, we set f'(x) = 0 and solve for x:
4x^3 - 64x = 0
Factor out 4x: 4x(x^2 - 16) = 0
Set each factor equal to zero: 4x = 0 or x^2 - 16 = 0
Solving these equations, we find x = 0, x = 4, and x = -4.
3. Next, we find the endpoints of the interval [-3, 9], which are -3 and 9. These are also critical points since they are the boundaries of our interval.
4. Now we evaluate the function f(x) at each of the critical points and endpoints to determine the function values at those points. Plug the values of x into the equation:
f(-3) = (-3)^4 - 32(-3)^2 + 2
= 81 - 288 + 2
= -205
f(0) = (0)^4 - 32(0)^2 + 2
= 0 - 0 + 2
= 2
f(4) = (4)^4 - 32(4)^2 + 2
= 256 - 512 + 2
= -254
f(9) = (9)^4 - 32(9)^2 + 2
= 6561 - 2592 + 2
= 3961
5. Finally, compare the function values obtained in step 4 to find the absolute minimum and maximum values in the given interval:
Absolute minimum: -254
Absolute maximum: 3961
Therefore, the absolute minimum value of the function f(x) is -254, and the absolute maximum value is 3961.