MHB Integration Properties: Get Help with Parts B & C

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The discussion focuses on a request for help with parts B and C of a quiz or test, following a successful attempt at part A. However, the forum's policy prohibits assistance with problems that impact grades. Users are encouraged to direct any concerns or questions via private message. The emphasis is on adhering to the rules set by the Math Help Boards. Overall, the conversation highlights the importance of academic integrity in seeking help.
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View attachment 4324 Please help me with this question. Part a was good but I don't understand parts b or c. Thanks in advanced.
 

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This appears to be part of an online quiz or test. It is MHB policy not knowingly to help with any problem that counts towards a user's grades, as per http://mathhelpboards.com/rules/. Please PM me if you have any thoughts or concerns or questions.
 

Thread 'Proving that convexity implies second order derivative being positive'

There are probably loads of proofs of this online, but I do not want to cheat. Here is my attempt: Convexity says that $$f(\lambda a + (1-\lambda)b) \leq \lambda f(a) + (1-\lambda) f(b)$$ $$f(b + \lambda(a-b)) \leq f(b) + \lambda (f(a) - f(b))$$ We know from the intermediate value theorem that there exists a ##c \in (b,a)## such that $$\frac{f(a) - f(b)}{a-b} = f'(c).$$ Hence $$f(b + \lambda(a-b)) \leq f(b) + \lambda (a - b) f'(c))$$ $$\frac{f(b + \lambda(a-b)) - f(b)}{\lambda(a-b)}...
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