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) = _________.

Answer 1

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,

$f\text{'}\left(x\right)=0$

$\Rightarrow 2\left(x+1\right)=0$

$\Rightarrow x+1=0$

$\Rightarrow x=-1$

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)___.