508F08Ex2solns

# Since both a and b are differentiable at t in the

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Since both A and B are differentiable at t 0 , in the limit as h 0 we find that M is differentiable at t 0 and M 0 ( t 0 ) = A 0 ( t 0 ) B ( t 0 ) + A ( t 0 ) B 0 ( t 0 ) .

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3 B–3. Let w ( x ) be a smooth function that satisfies w 00 - c ( x ) w = 0, where c ( x ) > 0 is a given function. a) Show that w cannot have a local positive maximum (that is, a maximum at an interior point where the function is positive). Also show that w cannot have a local negative minimum. Solution If w has a positive maximum at an interior point x 0 , then w 00 ( x 0 ) 0 and w ( x 0 ) > 0. Since c > 0, this gives w 00 ( x 0 ) - c ( x 0 ) w ( x 0 ) < 0, which is a contradiction. If w ( x ) has a local negative minimum, then the function - w ( x ) has a local positive maximum, which we have just shown cannot occur. b) [Uniqueness] If you also know that w (0) = a and w (1) = b , prove that there is at most one solution w ( x ) C 2 ([0 , 1]) with these properties. Solution Say there are two functions u ( x ) and v ( x ) that satisfy w 00 - c ( x ) w = 0 and have the same boundary conditions. Let z ( x ) := u ( x ) -

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