Maxima and Minima — Maths STD 12 Science — Question
Gujarat BoardEnglish MediumSTD 12 ScienceMathsMaxima and Minima4 Marks
Question
Determine two positive numbers whose sum is 15 and the sum of whose squares is maximum.
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Answer
Let the two positive numbers be x and y. Then, x + y = 15 ....(1) Now, z = x2 + y2 ⇒ z = x2 + (15 - x)2 [from eq.(1)] ⇒ z = x2 + x2 + 225 - 30x ⇒ z = 2x2 + 225 - 30x $\Rightarrow \frac{\text{dz}}{\text{dx}}=4\text{x}-30$ For maximum or minimum value of z, We must have $\Rightarrow \frac{\text{dz}}{\text{dx}}=0$ $\Rightarrow 4\text{x}-30=0$ $\Rightarrow \text{x}=\frac{15}{2}$ $\frac{\text{d}^{2}\text{z}}{\text{dx}^{2}}=4>0$ Substituting $\text{x}=\frac{15}{2}$ in (1), We get $ \text{y}=\frac{15}{2}$ Thus, z is minimum when $\text{x}=\text{y}=\frac{15}{2}$.
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