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Answer the following.
Determine the values of constants p and q such that the function f(x) = {psin x + q, x ≤ 0; x² + 2x…
Determine the values of constants $p$ and $q$ such that the function
$f(x) = \begin{cases} p\sin x + q, & x \le 0 \\ x^2 + 2x + 1, & x > 0 \end{cases}$
is differentiable at $x = 0$.
Answer
Answer
Official answer keyContinuity at $x = 0$: $\lim_{x\to 0^-}(p\sin x + q) = q$ and $\lim_{x\to 0^+}(x^2 + 2x + 1) = 1$, so $q = 1$.
Differentiability: LHD $= p\cos 0 = p$ and RHD $= 2(0) + 2 = 2$, so $p = 2$.
Final answer: $p = 2$, $q = 1$.