Question

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Answer 1

The given function is a "piecewise function, since there are 3 different functions, each valid only on its own interval. You'll need to differentiate 3 times. What are the domains of the resulting 3 derivatives?

Answer 2

ik the derivatives are 5, 2x, 4 but idk how to find the domains

Answer 3

First, take note that domain of polynomials are all the real numbers.
However, take note of the following limitations for a piecewise function.
Domain of derivatives of the respective sub-function of a piecewise function will not exceed the domain of its parent function.
In addition, at the end-points of each sub-function, if the value of the derivative on the left and that on the right do not match, then that end-point will not be part of the domain of the derivative.

Answer 4

x ≠ 4

Answer 5

For a function to be differentiable at a number x=c,
you must have the following:
\[f(c) \text{ exists } \\ f(c)=\lim_{x \rightarrow c^-} f(x) \text{ and } f(c)=\lim_{x \rightarrow c^+}f(x) \\ \text{ and you also must have } \\ \lim_{x \rightarrow c^-}f'(x)=\lim_{x \rightarrow c^+} f'(x)\]

so let's check the endpoints...
checking continuity first:
does 5x-6=x^2-2 as x approaches x=4?
does x^2-2=4x+10 as x approaches x=6?


if you answer yes to either we have a little more work to do

Answer 6

yeahh

Answer 7

You are given three different math'l expressions on 3 separate domains. You might want to graph those three math'l expressions on those domains, and then find the derivative of each of these 3 math'l expressions and graph the three separate derivatives below your first set of graphs.

This will help you towards deciding whether or not the derivatives have the same or different values where one subinterval ends and the next one begins.

Would you please graph the 3 separate component functions given in the original problem statement. Then, find the deriv. of each and graph each deriv. underneath.