Let g be the function given by g(x)=x^4−3x^3−x. What are all values of x such that g′(x)=12 ?
To find the values of x for which g'(x) = 12, we need to take the derivative of the function g(x) and set it equal to 12
To find the values of x for which g'(x) = 12, we need to take the derivative of the function g(x) and set it equal to 12.
First, let’s find g'(x) by taking the derivative of g(x) term by term:
g(x) = x^4 – 3x^3 – x
The derivative of x^4 is 4x^3.
The derivative of -3x^3 is -9x^2.
The derivative of -x is -1.
So, g'(x) = 4x^3 – 9x^2 – 1.
Now, we can set g'(x) equal to 12:
4x^3 – 9x^2 – 1 = 12
Rearranging the equation, we get:
4x^3 – 9x^2 – 13 = 0
This is a cubic equation that can be difficult to solve algebraically. One way to find the values of x is by using numerical methods or graphing the function to find the approximate solutions. However, it is not possible to find the exact values of x algebraically without the use of approximation methods.
If you need a particular set of values for x, I can help you solve the equation numerically or graphically. Alternatively, if you have any other questions or need further clarification, please let me know.
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