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Excel in math and science
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Extrema
How can you maximize your happiness under a budget? When does a function reach its minimum value? When does a curve change direction? The calculus of extrema explains these "extreme" situations.
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
Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.
Let \(P\) and \(Q\) be the two inflection points of the curve \(\displaystyle y=e^{-5x^2}.\) What is the area of the triangle \(OPQ,\) where \(O\) is the origin?
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
Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.
If \(\displaystyle{f(x)=\frac{1}{x^2+9}},\) the interval on which the curve \(y=f(x)\) is concave down can be expressed as \(a<x<b.\) What is the value of \(ab?\)
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
Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.
Let \(\displaystyle{f(x)=8+\frac{a}{x}+\frac{b}{x^2},}\) where \(a\) and \(b\) are constants. If the point \((-1,0)\) is the inflection point of the curve \(y=f(x),\) what is \(a+b?\)
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
Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.
If \( f(x) = x^{1/3}(x + 26) \), the interval on which the curve \( y = f(x) \) is concave down can be expressed as \( a < x < b \). What is the value of \( a + b \)?
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
Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.
Let \(y=(ax^2+30)e^x\) be the equation of a curve, where \(a\) is a constant. If \(A\) and \(B\) are two distinct arbitrary points on the curve and the arc between the two points always lies below the line connecting the two points, what is the range of \(a?\)