1) Suppose f is continuous on [1,3], f''(x) exists for all x in (1.3), f(1)=2, f(2)=-3 and f(3)=0. Show that there are points a,b in (1.3) such that f'(a)=0 and f''(b)4. 2) Suppose that f'(x) exists for all x in [a,b] and that f'(a)

Question

1)   Suppose f is continuous on [1,3], f''(x) exists for all x in (1.3), f(1)=2, f(2)=-3 and f(3)=0. Show that there are points a,b in (1.3) such that f'(a)=0 and f''(b)>4.
2) Suppose that f'(x) exists for all x in [a,b] and that f'(a)<f'(b). Take a number k in (f'(a),f'(b)). Let g(x)=f(x)-kx. Show that the function g takes its minimum value on [a,b] at some point c in (a,b) and deduce that f'(c)=k

anonymous · Answer

For 2) see the proof on http://en.wikipedia.org/wiki/Darboux_function#Darboux.27s_theorem

anonymous · Answer

For 1) there is a point a in [1,3] so that f attains its minimum. So
$ f(a) \leq -3 $ so a cannot be 1 or 3

anonymous · Answer

To finish one consider the function g(x)= f(x) - 4 x. g is continuous on [1,3] so it attains its minimum at a point c. There would be a b near c so that g'(b)>0, meaning that f'(b) >4. Show that b can be chosen different from 1 and 3

anonymous · Answer

thank you so much. but I couldn't understand the solution for question 1.

goformit100 · Answer

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