Respuesta :
Let's call the two number [tex] x [/tex] and [tex] y [/tex]. Since we know that the difference between them is [tex] 30 [/tex], if we assume [tex] x [/tex] to be the largest we have [tex] x-y=30 \iff x=y+30 [/tex]
Their product is [tex] xy [/tex], which we can write as
[tex](y+30)y = y^2+30y[/tex]This is the equation of a parabola, since it is a polynomial of degree [tex] 2 [/tex]. To find its minimum, simply derive it and set the derivative to zero: using the power rule
[tex]\frac{d}{dx}x^n = nx^{n-1}[/tex]
we have the following derivative:
[tex]2y+30 = 0 \iff y = -15[/tex]
Now that we have found one variable, we can substitute its value in the formula that related it to the other variable:
[tex] x = y+30 = -15+30 = 15 [/tex]
So, the two numbers are [tex] -15 [/tex] and [tex] 15 [/tex]
