The monthly demand equation for an electric utility company is estimated to be

p=58105x,

where p is measured in dollars and x is measured in thousands of killowatt-hours. The utility has fixed costs of

$5,000,000

per month and variable costs of

$32

per 1000 kilowatt-hours of electricity generated, so the cost function is

C(x)=5106+32x.

(a) Find the value of x and the corresponding price for 1000 kilowatt-hours that maximize the utility's profit.

(b) Suppose that the rising fuel costs increase the utility's variable costs from

$32

to

$46,

so its new cost function is

C1(x)=5106+46x.

Should the utility pass all this increase of

$14

per thousand kilowatt-hours on to the consumers?