MHB Understanding Integration Minus Integration in FTOC Proof

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The discussion centers on understanding the concept of "integration minus integration" within the context of FTOC Proof. The original poster expresses difficulty in grasping this concept. A hint is provided, leading to a resolution of the confusion. The poster acknowledges their understanding after receiving assistance. The conversation highlights the importance of clarification in complex mathematical topics.
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Hello MHB,
I am trying to understand FTOC Proof but I struggle on a integration minus integration, how do that works?

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Regards,
$$|\pi\rangle$$
 
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Hint :

$$-\int^x_a f(t)\, dt = \int^a_x f(t)\, dt$$
 
ZaidAlyafey said:
Hint :

$$-\int^x_a f(t)\, dt = \int^a_x f(t)\, dt$$
Got it now! Thanks for taking your time!:)

Regards,
$$|\pi\rangle$$
 

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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