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APPLICATION OF DERIVATIVES — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsAPPLICATION OF DERIVATIVES2 Marks
Question
Find the intervals in which the following function are strictly increasing or decreasing:
$10 - 6x - 2x^2$
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Answer

Given: $\text{f}\text{(x)} = 10-6\text{x} - 2\text{x}^2$ $\Rightarrow\ \text{f}'\text{(x)}=-6 -4\text{x} = -2(3 + 2\text{x)}\ \dots\text{(i)}$ $\text{Now} -2(3+2\text{x}) = 0\ \Rightarrow\ \text{x}= \frac{-3}{2}$
Therefore, we have two sub-intervals $\bigg(-\infty,\ \frac{-3} {2}\bigg)\text{and}\bigg(\frac{-3} {2},\ \infty\bigg)$. For interval $\bigg(- \infty,\ \frac{-3}{2}\bigg)$ taking x = -2 (say), from eq. (i), f'(x) = (-) (-) (+) > 0
Therefore, f is strictly increasing. For interval $ \bigg(\frac{-3}{2},\ \infty \bigg)$ taking x = -1 (say), from eq. (i), f'(x) = (-) (+) = (-) < 0Therefore, f is strictly decreasing.

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