Determine the values of constants and such that the function is differentiable at .

For $f(x)$ to be differentiable at $x = 0$, it must be both continuous and differentiable at $x = 0$. 1. Continuity at $x = 0$: $\text{LHL} = \lim_{x \to 0^-} (p\sin x + q) = q$. $\text{RHL} = \lim_{x \to 0^+} (x^2 + 2x + 1) = 1$. $f(0) = p\sin(0) + q = q$. For continuity, $\text{LHL} = \text{RHL} = f(0) \Rightarrow q = 1$. 2. Differentiability at $x = 0$: $\text{LHD} = \left.\frac{d}{dx}(p\sin x + q)\right|_{x=0} = \left. p\cos x \right|_{x=0} = p$. $\text{RHD} = \left.\frac{d}{dx}(x^2 + 2x + 1)\right|_{x=0} = \left. (2x + 2) \right|_{x=0} = 2$. For differentiability, $\text{LHD} = \text{RHD} \Rightarrow p = 2$. Thus, $p = 2$ and $q = 1$.