f ( x ) = 2 x 3 − 42 x 2 + 240 x + 11

Question

The function  has one local minimum and one local maximum.This function has a local minimum at  equals   

with value   

and a local maximum at  equals   

with value 

Answer 1

Because f(x) is an odd degree polynomial with a positive leading coefficient, its graph will begin in QIII (ie lim_{x→ - ∞} f(x) = - ∞) and end in QI (ie lim_x→∞ f(x) = ∞). This means the extremum with the smaller x-value will be the local max, and the extremum with the bigger x-value will be the local minimum. We can find both x-values by taking the derivative using power rule, setting it = 0, and solving:

f^'(x) = 6x² - 84x + 240 = 0

x² - 14x +40 = 0

(x - 10)(x - 4) = 0

x = 4 or x = 10

To find the y-values of these extrema, we can plug the values for x into f(x):

f(4) = 2·64 - 42·16 + 240·4 + 11 = 427 so local max at (4 , 427).

f(10) = 2·1,000 - 42·100 + 240·10 + 11 = 211 so local min at (10 , 211).

Answer 2

Use the first derivative test.