- #1
phillyolly
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Homework Statement
y=2√x-x
The Attempt at a Solution
First derivative:
-2x-2-1
Second derivative:
4x-3
4x-3=0
No solutions?
Mentallic said:But even so, if you correctly find the second derivative you'll see that it cannot equal 0, which means there is no inflection point. This means the graph of y=f(x) doesn't ever go from concave up to concave down or vice versa, it is always just one. Can you figure out whether it is always concave up or down without looking at the graph?
Bohrok said:[tex]y'' = -\frac{1}{2}x^{-\frac{3}{2}} = -\frac{1}{2x^{3/2}}[/tex]
x = 0 isn't in the domain because you would have 0 in the denominator, but that's not really important. This function y'' is never 0, it will never touch the x-axis, so y does not have an inflection point.
An inflection point is a point on a curve where the concavity changes. This means that the curve goes from being convex (curving upwards) to concave (curving downwards) or vice versa.
Identifying inflection points is important in many fields, including mathematics, physics, and economics. It helps us understand how a system or process is changing and can aid in making predictions or decisions.
To find inflection points, you must first find the second derivative of the function. Then, set the second derivative equal to zero and solve for the variable. The resulting value will be the x-coordinate of the inflection point.
Yes, a function can have multiple inflection points. This occurs when the concavity changes more than once on the curve.
Critical points are points where the first derivative of a function is equal to zero. Inflection points can occur at critical points, but not all critical points are inflection points. A function can have critical points without having any inflection points.