Let f be the function given by f(x) = 300x – x^3. On which of the following intervals is the function f increasing?
To determine on which intervals the function f(x) = 300x – x^3 is increasing, we need to find the values of x for which the derivative of f(x) is positive
To determine on which intervals the function f(x) = 300x – x^3 is increasing, we need to find the values of x for which the derivative of f(x) is positive.
The derivative of f(x) can be found by applying the power rule, which states that the derivative of x^n is n*x^(n-1), where n is a constant.
Applying the power rule, the derivative of f(x) = 300x – x^3 becomes:
f'(x) = 300 – 3x^2
To find the intervals where f(x) is increasing, we need to find the x-values such that f'(x) > 0.
Setting f'(x) > 0, we have:
300 – 3x^2 > 0
Rearranging and solving for x, we get:
3x^2 < 300 x^2 < 100 Taking the square root of both sides (remembering that we need to consider both positive and negative values for x), we get: x < 10 and x > -10
This tells us that the function f(x) = 300x – x^3 is increasing on the interval (-10, 10), which means that as x increases within this interval, the values of f(x) increase.
In summary, the function f(x) = 300x – x^3 is increasing on the interval (-10, 10).
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