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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 \( a+b? \)

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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 \(abm\)?

Excel in math and science

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It’s hard to learn from lectures and videos

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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 \( b-a? \)

Excel in math and science

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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 \(a+c\)?

Excel in math and science

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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 \( k^2 + p^2? \)

Excel in math and science

Master concepts by solving fun, challenging problems.

It’s hard to learn from lectures and videos

Learn more effectively through short, conceptual quizzes.

Our wiki is made for math and science

Master advanced concepts through explanations,
examples, and problems from the community.

Used and loved by 4 million people

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