– ) + x f ′ ( x ) = ( x – ) + x = ( x – ) ( x + x + ) Thus, f ′ ( x ) = gives x = or x + x + = for which there are no real roots. Also, there are no end points of the interval to be added to the set for which f ′ is zero, i.e., there is only one point, namely, x = . The value of f at this point is given by f ( ) = ( – ) + ( ) = . Thus, the distance between the solider and the helicopter is ( ) Note that is either a maximum value or a minimum value.
Since ( ) ( ) ( ) > it follows that is the minimum value of f x . Hence, is the minimum distance between the soldier and the helicopter. EXERCISE . .
Find the maximum and minimum values, if any, of the following functions given by (i) f ( x ) = ( x – ) + (ii) f ( x ) = x + x + (iii) f ( x ) = – ( x – ) + (iv) g ( x ) = x + . Find the maximum and minimum values, if any, of the following functions given by (i) f ( x ) = | x + | – (ii) g ( x ) = – | x + | + (iii) h ( x ) = sin( x ) + (iv) f ( x ) = |sin x + | (v) h ( x ) = x + , x ∈ (– , ) . Find the local maxima and local minima, if any, of the following functions. Find also the local maximum and the local minimum values, as the case may be: (i) f ( x ) = x