Answer :

Given:


Rate of working of an engine R, v is the speed of the engine:


, where 0<v<30


For finding the maximum/ minimum of given function, we can find it by differentiating it with v and then equating it to zero. This is because if the function f(x) has a maximum/minimum at a point c then f’(c) = 0.


Now, differentiating the function R with respect to v.




----- (1)


[Since and ]


Equating equation (1) to zero to find the critical value.





v2 = 400



v = 20 (or) v = -20


As given in the question 0<v<30, v = 20


Now, for checking if the value of R is maximum or minimum at v=20, we will perform the second differentiation and check the value of at the critical value v = 20.


Differentiating Equation (1) with respect to v again:





[Since and ]



----- (2)


Now find the value of



So, at critical point v = 20. The function R is at its minimum.


Hence, the function R is at its minimum at v = 20.


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