for the polynomial function ƒ(x)= -2x2 - 4x + 16, find all local and global extrema.

Question

for the polynomial function ƒ(x)= −2x2 − 4x + 16, find all local and global extrema.

Answer 1

Hi Olivia,

 

This can be done with elementary algebra or calculus. Since you used the term extrema, I am assuming you are working with calculus. To find the extrema, find the first derivative of the function

 

                                         f(x) = -2x² -4x +16  Therefore  f'(x) = -4x - 4.   We then set the first derivative equal to 0.

 

                                                       -4x - 4 = 0     and solve for x resulting in x = -1.  This is a critical point where a local minimum or local maximum is located. Substitute x = -1 into the function to find the value of the function at that point.    

                       f(-1) = -2(-1)² -4(-1) + 16 = 18  The point (-1,18) is a local minimum or maximum. To determine which it is we can use the second derivative test. Find the second derivative of f(x) which is the derivative of the first derivative. In this case    f''(x) = -4.  Since the value of the second derivative at -1 is -4, the graph is concave down which means it is a local maximum.