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APPLICATION OF DERIVATIVES — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsAPPLICATION OF DERIVATIVES5 Marks
Question
Find the absolute maximum and the absolute minimum value of the following functions in the given intervals:
$\text{f}(\text{x})=4\text{x}-\frac{\text{x}^{2}}{2}\ \text{in}\ [2,4,5]$
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Answer

We have, $\text{f}(\text{x})=4\text{x}-\frac{\text{x}^{2}}{2}$
$\Rightarrow\text{f}'(\text{x})=4-\text{x}$
For a local maximum or a local minimum value, We must have f'(x) = 0
$\Rightarrow4-\text{x}=0$
$\Rightarrow\text{x}=4$
Thus, the critical points of f are -2, 4 and 4.5.
Now, $\text{f}(-2)=4(-2)-\frac{(2)^{2}}{2}=-8-2=-10$
$\text{f}(4)=4(4)-\frac{(4)^{2}}{2}=16-8=8$
$\text{f}(4.5)=4(4.5)-\frac{(4.5)^{2}}{2}=18-10.125=7.875$
Hence, the absolute maximum value when x = 4 is 8 and the absolute minimum value when x = -2 is -10.

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