Maxima and Minima — Maths STD 12 Science — Question
Maharashtra BoardEnglish MediumSTD 12 ScienceMathsMaxima and Minima5 Marks
Question
Determine two positive numbers whose sum is $15$ and the sum of whose squares is maximum.
✓
Answer
Let the two positive numbers be x and y. Then,
$x + y = 15 ....(1)$
Now, $z = x^2 + y^2$
$\Rightarrow z = x^2 + (15 - x)^2 $[from eq.(1)]
$\Rightarrow z = x^2 + x^2 + 225 - 30x$
$\Rightarrow z = 2x^2 + 225 - 30x$
$\Rightarrow \frac{\text{dz}}{\text{dx}}=4\text{x}-30$
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 $\text{x}=\text{y}=\frac{15}{2}$.
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.