Application of Derivatives — Maths STD 12 Science — Question

According to the question, x + y = 24

$\Rightarrow y = 24 - x$ …(i)

And let z is the product of x and y.

$\Rightarrow $z = xy

$\Rightarrow$ z = x(24 - x) [From eq. (i)]

$\Rightarrow $z = 24x - x²

$\Rightarrow \frac{{dz}}{{dx}} = 24 - 2x$ and $\frac{{{d^2}z}}{{d{x^2}}} = - 2$ 

Now to find turning point, $\frac{{dz}}{{dx}} = 0$

$\Rightarrow 24 - 2x = 0 \Rightarrow x = 12$

At $x = 12,\frac{{{d^2}z}}{{d{x^2}}} = - 2$ [Negative]

$\therefore x = 12$ is a point of local maxima and z is maximum at ​​​​​​​x = 12.

$\therefore $  From eq. (i), ​​​​​​​ y = 24 - 12 = 12

Therefore, the two required numbers are 12 and 12.