I am very unsure of how to do this question..

Suppose f ''(x) < 0 for all x, and suppose f(1)< f(3)< f(2). Prove that f(0)< f(2) and f(4)< f(2).

Can somebody help me please?

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- Sep 6th 2009, 03:20 AM #1

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- Sep 6th 2009, 04:20 AM #2

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Well if the second derivative is always negative, the original function must be always concave, like an 'upside down' quadratic. Draw one of these and think what it could mean if the y-value at x=2 is higher than that at both x=1 and x=3. Try plotting roughly where these points would lie on the quadratic and then plot f(0) and f(4) relative to the ones you've already marked. Look at the heights of these points relative to the other ones.