(a)
Local minima: \(\frac{dy}{dx} = \sin(x + y) = 0\)
Solve: \(x + y = n\pi\), \(n \in \mathbb{Z}\)
For \(n = 0\): \(x + y = 0 \implies y = -x\)
Verify minima: \(\frac{d^2 y}{dx^2} = \cos(x + y) \cdot (1 + \frac{dy}{dx})\), at \(\sin(x + y) = 0\), \(\cos(x + y) = \pm 1\)
For \(x + y = 0\), \(\cos(0) = 1\), \(\frac{d^2 y}{dx^2} = 1 > 0\), confirms minima
Fits domain \(-4 \leq x \leq 5\), \(0 \leq y \leq 5\), and points like \((-1, 1)\), \((-3, 3)\)
Result: \(y = -x\) [3]

(b)
Local maxima: \(\sin(x + y) = 0 \implies x + y = n\pi\)
For \(n = 1\): \(x + y = \pi \implies y = -x + \pi\)
Verify maxima: For \(x + y = \pi\), \(\cos(\pi) = -1\), \(\frac{d^2 y}{dx^2} = -1 < 0\), confirms maxima
Fits domain and points like \((2, 1.14)\), \((0, 3)\), where \(1.14 \approx \pi – 2\), \(3 \approx \pi\)
Result: \(y = -x + \pi\) [2]