Help!!!!! D ??? Explain why or why not Determine whether the following statement

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Help!!!!! D ???

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. If f (c) = 0, then f has a local maximum or minimum at c. If fc) = 0, then f has an inflection point at c. F(x) = x^2 + 10 and G(x) = x^2 - 100 are antiderivatives of the same function. Between two local minima of a function continuous on (- infinity, infinity), there must be a local maximum. The linear approximation to f(c) = sin x at x = 0 is L(x) = x. If lim x rightarrow infinity f(x) = infinity and lim x rightarrow g(x) = infinity, then lim x rightarrow infinity (f(x) - g(x)) = 0

Explanation / Answer

for D)

suppose there isnt a local maximum btwn 2 local minimum.suppose x1 and x2 are local minima and f(x1)>f(x2)

then f(x)>f(x2) for x1<x<x2    since there is no absolute maximum, maximum should exist at end points in the interval i.e at x = x1 but x1 is absolute minima which contradicts our assumption that there is no absolute maxima

so there exists an absolute maximum betwn absolute minimas

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