Completing The Square
Consider two functions \[\begin{align} f(x) &= -4x^2 + 14 x + 4 \\ g(x) &= -7 x^2 + 8 x + 3. \end{align}\] Let \( a \) be the maximum value of \( f(x) + g(x) ,\) and \(b\) the value of \(x \) at that moment. Then what is the value of \( a+b? \)
For real numbers \(x\) and \(y\), the polynomial \[x^2+2y^2+2xy-8x-10y+30\] has a minimum value \(m\) at \(x=a\) and \(y=b\). What is the value of \(abm\)?
Let \(f(x)= 5 x + 60\) and \(g(x)= 4 x + 80.\) If \(a\) is the minimum value of \( f(x) \times g(x) \) and \(b\) is the value of \(x \) at that moment, what is the value of \( b-a? \)
Consider two points \(A=(2,5)\) and \(B=(4,4)\), and a third point \(P=(a, 0)\) on the \(x\)-axis. If the minimum value of \(\overline{PA}^2+\overline{PB}^2\) is \(c\), what is \(a+c\)?
Consider the parabola \[y = x^2 + k^4 + p^4 -2k^2 x -2p^2x + 2k^2p^2 + 36,\] where \(k\) and \(p\) are parameters. If the vertex of the parabola lies on the line \[ y = \frac{ 1 }{25} x, \] what is the value of \( k^2 + p^2? \)