The cost C of running a ship on a certain voyage is said to be given by c=av+b/v where v is the average speed in km/h and a,b are constant.find the value of v in terms of a and b to give the maximum cost,and also find the cost???

To find the value of v that gives the maximum cost, we need to find the critical points of the cost function and determine whether they correspond to a maximum or minimum. The cost function is given by c = av + b/v.

To find the critical points, we need to take the derivative of the cost function with respect to v and set it equal to zero. Let's do that:

c = av + b/v

Differentiating both sides of the equation with respect to v:

dc/dv = a - b/v^2

Next, we set the derivative equal to zero and solve for v:

a - b/v^2 = 0

b/v^2 = a

v^2 = b/a

v = ±√(b/a)

Since the average speed (v) cannot be negative in this context, we discard the negative square root and obtain:

v = √(b/a)

So, the value of v that gives the maximum cost is √(b/a).

To find the corresponding cost, we substitute this value of v back into the cost function:

c = a(√(b/a)) + b/√(b/a)

Simplifying further:

c = √(ab) + b/√(b/a)

We can rationalize the denominator:

c = √(ab) + b√(a/b)

Hence, the cost of running the ship with the value of v that gives the maximum cost is √(ab) + b√(a/b).

c = av + b/v

cv = av^2 + b
av^2 - cv + b = 0
now just use the quadratic formula to find v.

There will only be a maximum cost if a<0, since then the parabola will open downward.