Determining a slope and y-intercept from a piecewise function How would I begin solving a problem that is asking me to determine the value of $b$ such that $$f(x)= \begin{cases}3x^2+5&amp; x \le 2\\ mx+b &amp; x &gt; 2\end{cases}$$ is differentiable for all values of $x$?

Not asking for the answer here obviously, just the steps needed to get off the ground.
You must satisfy two requirements for differentiability at $x=2$:


$f(x)$ must be continuous at $x=2$.
The left limit of the derivative must equal the right limit of the derivative at $x = 2$.


That is,

$$\lim_{x\to 2^+}f(x)=\lim_{x\to 2^-}f(x)$$

and

$$\lim_{h\to 0^+}\frac{f(2+h)-f(2)}{h}=\lim_{h\to 0^-}\frac{f(2+h)-f(2)}{h}$$

You need to find values of $m$ and $b$ that satisfy these two equations. Do you think you can do it from here?
Hint: Differentiability at a point implies continuity at that point and that the derivatives are equal at that point.