Find the stationary points of the function (where the derivative is zero):
f(x) = ln(x ² + 7x + 14)
→ f '(x) = (2x + 7) / (x ² + 7x + 14)
The denominator is always positive, since
x ² + 7x + 14 = (x + 7/2)² + 7/4 ≥ 7/4 > 0
so f '(x) = 0 when
2x + 7 = 0 → x = -7/2
at which point f (-7/2) = ln(203/4) ≈ 3.927.
Also check the endpoints of the given domain:
f (-4) = ln(2) ≈ 0.693
f (1) = ln(22) ≈ 3.091
Then on the interval [-4, 1], we have max(f ) = ln(203/4) and min(f ) = ln(2).
