This is a problem from Optimization problem section Solution

(This is a problem from Optimization problem section

57:93 miles per hour

Let x be the speed of the truck in miles per hour. Then the duration of the trip is 300=x, the fuel

effiency is 4 -(50 - x)=10 = 9- x/10,

and the fuel expenditure is 300/(9 – x/10). The total cost C(x) of the

trip, then, is

C(x) = 1:19 300 /9 - x=10 + (27:50 + 11:33) 300/ x = 3570 /90 – x + 11649/x

The domain of C(x) is (0; 90), since those are the only values for which the fuel effciency expression is valid.

C d(x)= 3570 / (90 x)6^2 – 11649 /x^2

so the stationary points in the domain of C are only x = 57:93. There

are no endpoints and no places in the domain of C where C’’ does not exist.

The second derivative of C is

C’’(x) = 7140 /(90 x)^3 + 23298 /x^3 > 0

for all x in the domain of C. Since C is concave up on its entire domain,the stationary point must be a global minimum.