Taylor's theorem for multivariable functions

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For a multivariable function z=f(x,y), the total change can be derived from the individual changes in x and y, represented by Taylor's theorem. The discussion highlights the need for terms that account for simultaneous changes in x and y, specifically the mixed partial derivative ∂²f/∂x∂y. This term is crucial for accurately calculating the total change of z. The inquiry also references an external source for clarification on the derivation of the term 2fxy(x-a)(y-b). Understanding these concepts is essential for applying Taylor's theorem effectively in multivariable calculus.
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To get the total change you need terms that represent x and y changing together.
That leads to the term you asked about: ∂²f/∂x∂y
 
mathman said:
To get the total change you need terms that represent x and y changing together.
That leads to the term you asked about: ∂²f/∂x∂y

Thank you
 

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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