Answer :
Final answer:
To find where the function f(x) = x^(4)-12x^(3)-19 is decreasing, compute for the derivative f'(x) = 4x^(3)-36x^(2). Solve this for less than zero to get the values of x at which f(x) is decreasing.
Explanation:
To find the intervals where f(x) = x^(4)-12x^(3)-19 is decreasing, we first need to find the derivative of f(x), which represents the rate of change of the function. The derivative, f'(x) = 4x^(3)-36x^(2). This function gives us the slope of the tangent line to the curve at any point x, and we determine the intervals of decrease by finding where this derivative is less than zero.
Set 4x^(3)-36x^(2) less than zero, and solve for x. The solution gives us Ranges of x for which the original function f(x) is decreasing. If you have a graph of the function, these would be intervals on which the curve is going downhill when reading from left to right. Consider Figure 7.8, imagine a curve similar to f(x), and the decreasing part from X₁ to X₂.
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