Minimum Value of F(x): Does It Have a Maximum?

In summary, the minimum value of the function F(x) is determined by the minimum value of the function g(x) = x^2 - 2x, which occurs at x = 1. The function does not have a maximum value as its derivative never equals zero outside the interval 0 < x < 2.
  • #1
wonguyen1995
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0
Find the minimum value of F(x)=\int_{x^2-2x}^{0} 1/(1+t^2)\,d
Does it has a maximum value? why?
 
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  • #2
wonguyen1995 said:
Find the minimum value of F(x)=\int_{x^2-2x}^{0} 1/(1+t^2)\,d
Does it has a maximum value? why?

Defining the function as...

$\displaystyle f(x) = \int_{x^{2} - 2 x}^{0} \frac{d t } {1 + t^{2}}\ (1)$

... it is easy to see that is f(x)> 0 for 0 < x < 2 and f(x) < 0 outside this interval. Since the derivative of f(x) outside the interval 0 < x < 2 never vanishes, f(x) don't have minimum. The maximum of the function is the minimum of the function $\displaystyle g(x) = x^{2} - 2\ x$ and it happens for x=1...

Kind regards

$\chi$ $\sigma$
 
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FAQ: Minimum Value of F(x): Does It Have a Maximum?

What is the minimum value of a function?

The minimum value of a function is the smallest output value that the function can produce. It is the point on the graph of the function where the y-value is lowest.

How is the minimum value of a function determined?

The minimum value of a function can be determined by finding the critical points of the function, which are points where the slope of the curve is zero. These points can be found by taking the derivative of the function and setting it equal to zero. The smallest output value among these critical points is the minimum value of the function.

Can a function have more than one minimum value?

Yes, a function can have more than one minimum value. This can occur if the function has multiple local minimum points, which are points on the graph where the function is at a low point but not the absolute lowest point. However, a function can only have one absolute minimum value.

Is the minimum value of a function always the same as the maximum value?

No, the minimum value and maximum value of a function are not always the same. The maximum value of a function is the highest output value that the function can produce, and it is the point on the graph where the y-value is highest. These values can be the same in some cases, but it is not always the case.

Can a function have a minimum value if it does not have a maximum value?

Yes, a function can have a minimum value even if it does not have a maximum value. This can occur if the function continues to decrease indefinitely, never reaching a maximum point. In this case, the minimum value would be the lowest possible output value for the function.

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