6. Find both the maximum profit that a company can make, if the profit function is given by p(x) = 41 - 72x - 18x^2. 7. Find both the maximum value and the minimum value of 3x^4 -... 6. Find both the maximum profit that a company can make, if the profit function is given by p(x) = 41 - 72x - 18x^2. 7. Find both the maximum value and the minimum value of 3x^4 - 8x^3 + 12x^2 - 48x + 25 on the interval [0, 3]. 8. At what points in the interval [0, 2π] does the function sin 2x attain its maximum value? 9. What is the maximum value of the function sin x + cos x? 10. Find the maximum value of 2x^3 - 24x + 107 in the interval [1, 3]. Find the maximum value of the same function in [-3, -1].
Understand the Problem
The questions are asking for the maximum and minimum values of various mathematical functions and the points where these extrema occur within specified intervals.
Answer
Maximum value: $89$, Minimum value: $75$
Answer for screen readers
Maximum value: $89$
Minimum value: $75$
Steps to Solve
- Finding Derivatives To find the maximum and minimum values of the function ( f(x) = 2x^3 - 24x + 107 ), we first take the derivative:
$$ f'(x) = 6x^2 - 24 $$
- Setting the Derivative to Zero Next, we set the derivative equal to zero to find critical points:
$$ 6x^2 - 24 = 0 $$
- Solving for Critical Points Solving for ( x ):
$$ 6x^2 = 24 $$
$$ x^2 = 4 $$
$$ x = 2 \text{ or } x = -2 $$
- Evaluating Endpoints and Critical Points Now, we will evaluate the function ( f(x) ) at the endpoints ( x = 1 ) and ( x = 3 ), as well as the critical points.
- For ( x = 1 ):
$$ f(1) = 2(1)^3 - 24(1) + 107 = 2 - 24 + 107 = 85 $$
- For ( x = 2 ):
$$ f(2) = 2(2)^3 - 24(2) + 107 = 16 - 48 + 107 = 75 $$
- For ( x = 3 ):
$$ f(3) = 2(3)^3 - 24(3) + 107 = 54 - 72 + 107 = 89 $$
- Comparing Values Finally, we compare the values obtained at the endpoints and critical points:
- ( f(1) = 85 )
- ( f(2) = 75 )
- ( f(3) = 89 )
The maximum value is ( 89 ) and the minimum value is ( 75 ).
Maximum value: $89$
Minimum value: $75$
More Information
This function is a cubic polynomial, and polynomial functions are continuous and differentiable over all real numbers. The maximum and minimum values can be found by examining critical points and endpoints within the given interval.
Tips
- Forgetting to check endpoints along with critical points can lead to missing the maximum or minimum.
- Failing to simplify derivative equations correctly, which can result in incorrect critical points.
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