How Do You Determine Values of n and m for Differentiability?

[noname1]

please delete thread, i made a mistake

 

[Mark44]

for what values of n & m will be differentiable for all values of x

nx+m, x<=-1
nx^3+x+2m, X>-1


i have as n = to 0 and m = to 1 is this correct?

Your question doesn't make much sense the way it's written, or maybe I don't understand what you're trying to say.

I'm assuming that m and n are real numbers.

nx + m is differentiable for any values of m and n, regardless of the value of x. If the domain is restricted to x <= -1, it's still differentiable.

nx³ +x + 2m is differentiable for all values of m and n, regardless of the value of x.

It occurs to me that what you have, but didn't explain very well, is a function whose formula is different on the two intervals. If that's the case, you want to find m and n so that
$$1) \lim_{x \rightarrow -1^-} f(x)~=~\lim_{x \rightarrow -1^+} f(x)$$
$$2) \lim_{x \rightarrow -1^+} f'(x)~=~\lim_{x \rightarrow -1^+} f'(x)$$
If the first condition is satisfied, your function will be continuous for all x. If the second is satisfied, the derivative will be continuous for all x.

 

[noname1]

that how the question is written

i have to find the value of n and m that will make them differentiable, i solve the first one and it give me m = -1+n,

than i do limit from the right side is equal to the left resolve it and n = 1

than i replace n from the initial equation and than gives me 0, is this correct?