Find the intervals in which the function f given by f(x) = 2x^2 -… - Vidyadip

Question

Find the intervals in which the function f given by $f(x) = 2x^2 - 3x$ is (a) strictly increasing, (b) strictly decreasin.

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Answer

Given: $\text{f}\text{(x)} = 2x^2 - 3\text{x }\Rightarrow \text{f}'\text{(x)} = 4\text{x}-3\ \dots\text{(i)}$
$\text{Now}\ 4\text{x}-3=0\ \Rightarrow \ \text{x}=\frac{3}{4}$
Therefore, we have two intervals $\bigg(-\infty,\frac{3}{4}\bigg)\text{ and } \bigg(\frac{3}{4},\infty \bigg).$

For interval $\bigg(\frac{3}{4},\infty\bigg),$ taking x = 1, (say), then from eq. (i), f'(x) > 0.

Therefore, f is strictly increasing in $\bigg(\frac{3}{4},\infty\bigg).$

For interval $\bigg(-\infty,\frac{3}{4}\bigg),$ taking x = 0.5, (say), then from eq. (i), f'(x) < 0.

Therefore, f is strictly decreasing in $\bigg(-\infty,\frac{3}{4}\bigg).$

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