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Question

# If m and M respectively denote the minimum and maximum values of f(x) = (x + 1)2 + 3 in the interval [−3, 1], then the ordered pair (m, M) = _________.

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Solution

## The given function is f(x) = (x + 1)2 + 3, x ∈ [−3, 1]. f(x) = (x + 1)2 + 3 Differentiating both sides with respect to x, we get $f\text{'}\left(x\right)=2\left(x+1\right)$ For maxima or minima, $f\text{'}\left(x\right)=0$ $⇒2\left(x+1\right)=0$ $⇒x+1=0$ $⇒x=-1$ Now, $f\text{'}\text{'}\left(x\right)=2>0$ So, x = −1 is the point of local minimum of f(x). At x = −1, we have f(−1) = (−1 + 1)2 + 3 = 0 + 3 = 3 At x = −3, we have f(−3) = (−3 + 1)2 + 3 = 4 + 3 = 7 At x = 1, we have f(1) = (1 + 1)2 + 3 = 4 + 3 = 7 Thus, the minimum value of f(x) is 3 and the maximum value of f(x) is 7. ∴ m = 3 and M = 7 Thus, the ordered pair (m, M) is (3, 7). If m and M respectively denote the minimum and maximum values of f(x) = (x + 1)2 + 3 in the interval [−3, 1], then the ordered pair (m, M) = ___(3, 7)___.

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