1. Find values of the constants $k$ and $m$, if possible, that will make the function $f(x)$ continuous everywhere.
\[
f(x)=\left\{\begin{array}{ll}
x^{2}+5, & x> 2 \\
m(x+1)+k, & -1< x \leq 2 \\
2 x^{3}+x+7, & x \leq-1
\end{array}\right.
\]

Answer

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Answer

$\boxed{k = 4, m = \frac{5}{3}}$

Steps

Step 1 ：Find the limit of the function as x approaches -1 from the left and from the right, and equate them: $2(-1)^3 + (-1) + 7 = m(-1+1) + k$

Step 2 ：Find the limit of the function as x approaches 2 from the left and from the right, and equate them: $m(2+1) + k = 2^2 + 5$

Step 3 ：Solve the system of equations to find the values of k and m: $k = 4$ and $m = \frac{5}{3}$

Step 4 ：$\boxed{k = 4, m = \frac{5}{3}}$