Question
Find the intervals in which the following function are strictly increasing or decreasing:
$x^2 + 2x - 5$
$x^2 + 2x - 5$
✓
Answer
Given: $\text{f} \text{(x)} = \text{x}^2 + 2\text{x} - 5$$\Rightarrow\ \text{f'}\text{(x)} = 2\text{x} + 2 = 2\text{(x} + 1)\ \dots\text{(i)}$
$\text{Now } 2(\text{x} +1) = 0 \ \Rightarrow\ \text{x} = -1$
Therefore, we have two sub-intervals $(-\infty,\ -1)\text{ and }( -1,\ \infty).$
For interval $(- \infty,\ -1)$ taking x = -2 (say), from eq. (i), f'(x) = (-) < 0
Therefore, f is strictly decreasing.
For interval $(-1, \ \infty)$ taking x = 0 (say), from eq. (i). f'(x) = (+) > 0
Therefore, f is strictly increasing.
$\text{Now } 2(\text{x} +1) = 0 \ \Rightarrow\ \text{x} = -1$
Therefore, we have two sub-intervals $(-\infty,\ -1)\text{ and }( -1,\ \infty).$
For interval $(- \infty,\ -1)$ taking x = -2 (say), from eq. (i), f'(x) = (-) < 0
Therefore, f is strictly decreasing.
For interval $(-1, \ \infty)$ taking x = 0 (say), from eq. (i). f'(x) = (+) > 0
Therefore, f is strictly increasing.
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