For the real numbers \(a\) \(=\) 24 and \(b\) \(=\) 62, if \(a\) is increased by \(u\) \(=\) \(1\) and \(b\) is decreased by \(v\) \(=\) \(6\), then show that the increment in product is always equal to \(ub\) \(-\) \(av\) \(-\) \(uv\).
Proof:
Given, \(a\) \(=\) 24, \(b\) \(=\) 62, \(u\) \(=\) \(1\), and \(v\) \(=\) \(6\).
Their product \(ab\) \(=\) .
By the given condition, the new product is given by .
The value of the new product \(=\)
The difference in products \(=\)
By the distributive property, the increment in product is given by .
The increment in product using the obtained property \(=\) .
Difference in product Increment in product
Hence, proved.