9514 1404 393
Answer:
- absolute maximum: (3, 27e^-3) ≈ (3, 1.24425)
- absolute minimum: (-4, -64e^4) ≈ (-4, -3494.28)
Step-by-step explanation:
The absolute extremes will be at one of two locations:
- where the derivative is zero
- at an end of the interval
Here, the absolute maximum occurs where the derivative is zero (and the second derivative is negative).
f'(x) = 3x^2·e^-x -x^3·e^-x = x^2·e^-x·(3 -x) ⇒ f'(3) = 0
The absolute minimum occurs at the left end of the interval, where both x^3 and e^-x have large magnitude.
The minimum is ...
(-4)^3×e^-(-4) ≈ -3494.21 . . . . six significant figures
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A graph can show you the function tends toward -∞ as x goes to large negative values. Hence, the absolute minimum on an interval beginning in the left half-plane will be at the left end of the interval.
