Answer
No.
Work Step by Step
Step 1. To determine if the piecewise defined function is differentiable at the origin, we need to make sure that the function is continuous at the origin and the left and right derivatives are equal.
Step 2. Right side: $\lim_{x\to0^+}f(x)=2(0)-1=-1$
Step 3. Left side: $\lim_{x\to0^-}f(x)=0^2+2(0)+7=7$
Step 4. Since the left and right limits are not equal, the function is not differentiable at the origin.
