how do you find the absolute maximum of 2x^3+3x^2-12x-7 on interval (-3,0)
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To find the absolute maximum of a function on a given interval, you can follow these steps:
1. Find the critical points within the interval (-3, 0) by taking the derivative of the function and setting it equal to zero.
- In this case, the derivative of the function f(x) = 2x^3 + 3x^2 - 12x - 7 is f'(x) = 6x^2 + 6x - 12.
2. Solve the equation f'(x) = 0 to find the critical points. In this case, we have:
6x^2 + 6x - 12 = 0
You can divide the equation by 6 to simplify it:
x^2 + x - 2 = 0
Factor the equation:
(x + 2)(x - 1) = 0
Set each factor equal to zero and solve for x:
x + 2 = 0 --> x = -2
x - 1 = 0 --> x = 1
So, the critical points within the interval (-3, 0) are x = -2 and x = 1.
3. Evaluate the function at the critical points and at the endpoints of the interval (-3, 0).
- Plug in the values of -3, -2, 0, and 1 into the original function f(x) = 2x^3 + 3x^2 - 12x - 7 to find the corresponding function values.
4. Compare the function values obtained in step 3 to determine the absolute maximum.
- The largest function value will be the absolute maximum.
In this case, follow the steps above and compare the function values at -3, -2, 0, and 1 to find the absolute maximum.