Answer to Question #268313 in Calculus for toodles

Question

(1) Find and classify the extremes value of f(x) = 6x⁴ - 4x⁶ over the interval [ -2, 2].

(2) A retailer determines that the cost of ordering and storing units of a product can be modelled by C(x) = 3x + (2000/x), 0 ≤ x ≤ 200. Find the order size that will minimize ordering and storage cost. 

Answer 1

2)

$C'(x) = 3x + (2000/x)=3-2000/x^2=0$

order size that will minimize ordering and storage cost:

$x=\sqrt{2000/3}=25.81$ units

1)

$f(-2)=f(2) = 6x^4 - 4x^6=-160$

$f'(x)=24x^3-24x^5=0$

$x_1=0,x_2=1,x_3=-1$

$f(0)=0,f(1)=f(-1)=2$

so, there is absolute minimum at points (-2,-160) and (2,-160)

and absolute maximum at points (-1, 2) and (1, 2)

at point (0,0) derivative change sign from - to +, so it is local minimum