- #1
phymatter
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what is the reason that the minima and maxima of ( ex2 -1 )1/2 and ( ex2 -1 ) are the same ??
Why Do (ex2-1)1/2 and (ex2-1) Have the Same Minima/Maxima?
- Thread starter phymatter
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In summary, the reason that the minima and maxima of (ex2 - 1)1/2 and (ex2 - 1) are the same is because of the property of monotone increasing functions. This means that for a function g, if the maxima and minima (and the local maxima and minima) of g(f(x)) and f(x) will be the same. This property can be proven through mathematical reasoning.
- #2
TylerH
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How do you find the max and min of a function?
- #3
tiny-tim
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hi phymatter! 
this has nothing to do with calculus …
if g is a monotone increasing function, then the maxima and minima (and the local maxima and minima) of g(f(x)) and f(x) will be the same …
this has nothing to do with calculus …
if g is a monotone increasing function, then the maxima and minima (and the local maxima and minima) of g(f(x)) and f(x) will be the same …
now prove it!
- #4
phymatter
- 131
- 0
tiny-tim said:hi phymatter!
this has nothing to do with calculus …
if g is a monotone increasing function, then the maxima and minima (and the local maxima and minima) of g(f(x)) and f(x) will be the same …
now prove it!
thanks tiny-tim! :)
1. Why do (ex2-1)1/2 and (ex2-1) have the same minima and maxima?
The reason for this is because both functions share the same exponent of 2-1. When the exponent is raised to a power of 1/2, it results in the square root of the function, which has the same minima and maxima as the original function.
2. Is there a mathematical explanation for why (ex2-1)1/2 and (ex2-1) have the same minima and maxima?
Yes, there is a mathematical explanation for this. When we take the derivative of both functions, we get the same result because the exponent of 2-1 is a constant. And when we set the derivative equal to 0 to find the minima and maxima, we get the same values for both functions.
3. Can you give an example to illustrate why (ex2-1)1/2 and (ex2-1) have the same minima and maxima?
Sure, let's take the functions f(x) = (x2-1)1/2 and g(x) = x2-1. When we plot these functions on a graph, we can see that they have the same minima and maxima at x = 0 and x = 2. This is because both functions share the same exponent of 2-1, resulting in the same values when taking the square root.
4. How does the symmetry of (ex2-1)1/2 and (ex2-1) contribute to them having the same minima and maxima?
The symmetry of both functions plays a crucial role in them having the same minima and maxima. When we reflect the graph of (ex2-1)1/2 over the y-axis, it becomes the same as the graph of (ex2-1). This symmetry ensures that both functions have the same values at corresponding points, leading to the same minima and maxima.
5. Is there a practical application for understanding why (ex2-1)1/2 and (ex2-1) have the same minima and maxima?
Yes, understanding this concept can be useful in various fields such as physics, engineering, and economics. It allows us to easily find the minima and maxima of functions with shared exponents, which can be helpful in optimizing systems or predicting trends. For example, in economics, this concept can be applied to analyze demand and supply curves with the same exponent.
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