Step-by-Step Deconstruction

Topic: Monotonicity & Extrema in AOD

(i) Monotonicity Analysis (Increasing/Decreasing)

1. Differentiate f(x) using Product Rule:
f(x) = e^x sin x
f'(x) = e^x cos x + e^x sin x = e^x(sin x + cos x)

2. Find Critical Points for x ∈ [0, π]:
Set f'(x) = 0 → e^x(sin x + cos x) = 0
Since e^x ≠ 0, sin x + cos x = 0 → tan x = -1
In [0, π], x = 3π/4.

3. Test Intervals:
• Increasing: In [0, 3π/4), f'(x) > 0. f(x) is Increasing.
• Decreasing: In (3π/4, π], f'(x) < 0. f(x) is Decreasing.

(ii) Testing the Critical Point (x = 3π/4)

Using the Second Derivative Test:
f”(x) = d/dx [e^x(sin x + cos x)]
f”(x) = e^x(sin x + cos x) + e^x(cos x – sin x)
f”(x) = 2e^x cos x

Evaluate at x = 3π/4:
f”(3π/4) = 2e^3π/4 cos(3π/4)
Since cos(3π/4) = -1/√2 (Negative), f”(3π/4) < 0.

Conclusion: x = 3π/4 is a point of LOCAL MAXIMUM.

⚠️ Silly Mistake Audit: The “Product Rule” Slip

Most students fail at the very first step by forgetting to apply the Product Rule [u’v + uv’] to the function e^xsin(x). Forgetting the e^xcos(x) term leads to incorrect critical points and flawed interval analysis. Always double-check your initial derivative before testing monotonicity!