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kodGreya [7K]
3 years ago
13

Investigate the existence of a extrema value, what does the markered part mean?

Mathematics
1 answer:
OLga [1]3 years ago
5 0

Answer: We have

f'(x) = a x + b,

f'(x) = 0 at x = -b/a

f(x) = a x^2 / 2 + b x + c


Meaning of marked part


❟ ∵ a<0 ❟ f is a quadratic function

∴ f has absolute maximum value at x = -b/a


For all a with a less than zero, f is a quadratic function. Therefore f has a global maximum at x = -b/a


That typesetting seems very sloppy. It probably is supposed to be


∀a < 0, f is a quadratic function.


The second sentence is sloppy in use of "absolute". It can't mean absolute value, so presumably it means "global".


Sometimes a minimum or maximum is only local, but a quadratic function has exactly one extrema, and it is global. And if a < 0, the extrema is a global maximum.


Step-by-step explanation:


An extrema (minimum or maximum) for f(x) occurs only where f'(x) = 0, that is, when the slope of the tangent at x is zero.


But if the function crosses its tangent at that point, the point is an inflection point, not an extrema. A quadratic never crosses it's tangent.


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