How to Find the Maximum of the Function f(x)=12x-32+24√(9-3x), for x≤3?

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Thread starter Albert1
Start date Apr 29, 2016
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In summary, a maximum in mathematics is the highest value of a function or a set of data points. It can be found using methods such as graphing, differentiation, or optimization algorithms. Finding the maximum is important in various fields and a function can have multiple maximum values. To determine if a critical point is a maximum or a minimum, the second derivative test or the first derivative test can be used.

Apr 29, 2016

Albert1

$f(x)=12x-32+24\sqrt {9-3x}, x\leq 3$
find $max f(x)$

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Apr 29, 2016

kaliprasad

Gold Member

MHB

Albert said:

$f(x)=12x-32+24\sqrt {9-3x}, x\leq 3$
find $max f(x)$

let 9-3x = y^2 so 3x = 9-y^2 so $f(x) = 36-4y^2 - 32 + 24y = 4 - (4y^2 - 24 y) = 4 - 4(y- 3)^2 + 36 = 40 - 4(y-3)^2$
lowest when y = 3 ( range of y is >0 feasible and x = 0 so i we get 40 as maximum $f(x)$

Related to How to Find the Maximum of the Function f(x)=12x-32+24√(9-3x), for x≤3?

1. What is the definition of a maximum in mathematics?

A maximum in mathematics refers to the highest value of a function or a set of data points. It is the point at which the function reaches its peak or the data reaches its highest value.

2. How do I find the maximum of a function?

To find the maximum of a function, you can use various methods such as graphing, differentiation, or optimization algorithms. Graphing involves plotting the function and visually identifying the highest point. Differentiation involves finding the derivative of the function and setting it to zero to find the critical points. Optimization algorithms use numerical methods to find the maximum value.

3. What is the importance of finding the maximum of a function?

Finding the maximum of a function is important in various fields such as economics, engineering, and science. It helps in identifying the peak value of a variable, which can be useful in decision-making and optimization problems. It also helps in understanding the behavior of a function and its relationship with other variables.

4. Can a function have more than one maximum value?

Yes, a function can have multiple maximum values. These are known as local maximums and occur at different points on the function. However, a function can only have one absolute maximum, which is the highest point on the entire function.

5. How can I determine if a critical point is a maximum or a minimum?

To determine if a critical point is a maximum or a minimum, you can use the second derivative test or the first derivative test. The second derivative test involves evaluating the second derivative at the critical point. If the second derivative is positive, the critical point is a minimum, and if it is negative, it is a maximum. The first derivative test involves evaluating the first derivative on either side of the critical point. If the derivative changes from positive to negative, the critical point is a maximum, and if it changes from negative to positive, it is a minimum.