Question:

Step 1: Set up continuity condition at $x = -\frac{\pi}{2}$ For continuity at $x = -\frac{\pi}{2}$, the left-hand limit, right-hand limit, and function value must be equal: $$ \lim_{x \to -\frac{\pi}{2}^-} f(x) = \lim_{x \to -\frac{\pi}{2}^+} f(x) = f\left(-\frac{\pi}{2}\right) $$ Step 2: Evaluate left-hand limit at $x = -\frac{\pi}{2}$ Using $f(x) = -4\sin x + \cos x$ for $x \leq -\frac{\pi}{2}$: $$ \lim_{x \to -\frac{\pi}{2}^-} f(x) = -4\sin\left(-\frac{\pi}{2}\right) + \cos\left(-\frac{\pi}{2}\right) = -4(-1) + 0 = 4 $$ Step 3: Evaluate right-hand limit at $x = -\frac{\pi}{2}$ Using $f(x) = a \sin x + b$ for $-\frac{\pi}{2} < x < \frac{\pi}{2}$: $$ \lim_{x \to -\frac{\pi}{2}^+} f(x) = a \sin\left(-\frac{\pi}{2}\right) + b = a(-1) + b = -a + b $$ Step 4: Apply continuity at $x = -\frac{\pi}{2}$ For continuity, we equate the limits: $$ 4 = -a + b \quad \text{(Equation 1)} $$ Step 5: Set up continuity condition at $x = \frac{\pi}{2}$ For continuity at $x = \frac{\pi}{2}$, the left-hand limit and right-hand limit must be equal: $$ \lim_{x \to \frac{\pi}{2}^-} f(x) = \lim_{x \to \frac{\pi}{2}^+} f(x) $$ Step 6: Evaluate left-hand limit at $x = \frac{\pi}{2}$ Using $f(x) = a \sin x + b$ for $-\frac{\pi}{2} < x < \frac{\pi}{2}$: $$ \lim_{x \to \frac{\pi}{2}^-} f(x) = a \sin\left(\frac{\pi}{2}\right) + b = a(1) + b = a + b $$ Step 7: Evaluate right-hand limit at $x = \frac{\pi}{2}$ Using $f(x) = \cos x + 2$ for $x \geq \frac{\pi}{2}$: $$ \lim_{x \to \frac{\pi}{2}^+} f(x) = \cos\left(\frac{\pi}{2}\right) + 2 = 0 + 2 = 2 $$ Step 8: Apply continuity at $x = \frac{\pi}{2}$ For continuity, we equate the limits: $$ a + b = 2 \quad \text{(Equation 2)} $$ Step 9: Solve the system of equations We have the following system of linear equations: 1. $-a + b = 4$ 2. $a + b = 2$ Adding Equation 1 and Equation 2: $$ (-a + b) + (a + b) = 4 + 2 $$ $$ 2b = 6 $$ $$ b = 3 $$ Substitute $b=3$ into Equation 2: $$ a + 3 = 2 $$ $$ a = 2 - 3 $$ $$ a = -1 $$ Step 10: State the solution The values for $a$ and $b$ that ensure the function $f(x)$ is continuous are $a = -1$ and $b = 3$.