Question

If the function given by
$f(x) = -2 \sin x$ for $-\pi \le x < -\frac{\pi}{2}$
$f(x) = a \sin x + b$ for $-\frac{\pi}{2} < x < \frac{\pi}{2}$
$f(x) = \cos x$ for $\frac{\pi}{2} \le x \le \pi$
is continuous in $[-\pi, \pi]$, then the value of $(3a + 2b)^3$ is

• A. $1$
• B. $8$
• C. $-1$
• D. $-8$

Hint:

Whenever a problem states a piecewise function is continuous and gives a relation like $a + b = 0$, you instantly know $a$ and $b$ are opposites! Substituting $a = -b$ early into your first equation turns it into a simple one-variable solution in a flash.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
The problem presents a piecewise defined function over the interval $[-\pi, \pi]$. Since the function is stated to be continuous throughout this entire domain, it must strictly be continuous at the transition boundary joints $x = -\frac{\pi}{2}$ and $x = \frac{\pi}{2}$. We need to find the parameters $a$ and $b$ to compute the target value.

Step 2: Key Formula or Approach:
For a function to be continuous at an interior point $x = c$, the Left-Hand Limit (LHL), Right-Hand Limit (RHL), and the value of the function at that point must all be equal:
$$\lim_{x \to c^-} f(x) = \lim_{x \to c^2} f(x) = f(c)$$

Step 3: Detailed Explanation:
1. Continuity at $x = -\frac{\pi}{2}$:
$$\lim_{x \to -\frac{\pi}{2}^-} f(x) = \lim_{x \to -\frac{\pi}{2}^+} f(x)$$ $$-2 \sin\left(-\frac{\pi}{2}\right) = a \sin\left(-\frac{\pi}{2}\right) + b$$ Since $\sin\left(-\frac{\pi}{2}\right) = -1$:
$$-2(-1) = a(-1) + b \implies 2 = -a + b \quad \dots \text{(Equation 1)}$$ 2. Continuity at $x = \frac{\pi}{2}$:
$$\lim_{x \to \frac{\pi}{2}^-} f(x) = \lim_{x \to \frac{\pi}{2}^+} f(x)$$ $$a \sin\left(\frac{\pi}{2}\right) + b = \cos\left(\frac{\pi}{2}\right)$$ Since $\sin\left(\frac{\pi}{2}\right) = 1$ and $\cos\left(\frac{\pi}{2}\right) = 0$:
$$a(1) + b = 0 \implies a + b = 0 \implies a = -b \quad \dots \text{(Equation 2)}$$ 3. Solve the simultaneous equations:
Substitute Equation 2 ($a = -b$) into Equation 1:
$$2 = -(-b) + b \implies 2 = 2b \implies b = 1$$ Since $a = -b$, we find $a = -1$.
4. Evaluate the requested numerical expression:
$$(3a + 2b)^3 = (3(-1) + 2(1))^3 = (-3 + 2)^3 = (-1)^3 = -1$$ This calculation matches option (C).

Step 4: Final Answer:
The value of the expression is $-1$, which corresponds to option (C).