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प्रश्न
The interval in which y = x2 e–x is increasing is ______.
विकल्प
(– ∞, ∞)
(– 2, 0)
(2, ∞)
(0, 2)
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उत्तर
The interval in which y = x2 e–x is increasing is (0, 2).
Explanation:
x2 - e-x
`dy/dx = 2xe^-x - x^2 e^-x`
= xe-x (2 - x)
If f'(x) = 0
xe-x (2 - x) = 0
x = 0, 2
x = 0 and x = 2 divide the real line into intervals `(- infty, 0), (0, 2)` and `(2, infty)`.
Thus, `(- infty, -1)` and `(1, infty)` represent the intervals.
The function y is continuously increasing in the interval (0, 2).
| Interval | (- ∞, 0) | (0, 2) | (2, ∞ ) |
| Sign of x | -ve | +ve | +ve |
| sign of (2 - x) | +ve | +ve | -ve |
| sign of e-x | +ve | +ve | +ve |
| sign of f' (x) | -ve | +ve | -ve |
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