31) Locate, name, and classify the extrema of the function. Determine intervals which the function is increasing and decreasing
f(x) = -x(x^2 - 2)
32) Determine the end behavior of the function
f(x) = 2x - x^3
#31
since f'(x) = -3x^2+2
the extrema are at x = ±√(2/3)
f' > 0 for -√(2/3) x < √(2/3) so f is decreasing on that interval, and increasing elsewhere.
#32
Since f(x) is odd, and the x^3 coefficient is -1, it rises to the left and drops to the right.
To locate, name, and classify the extrema of the function f(x) = -x(x^2 - 2), we first need to find the critical points where the derivative of the function is equal to zero or undefined.
To find the derivative of f(x), we use the product rule:
f'(x) = -x(3x^2 - 2) + (-1)(x^2 - 2)
= -3x^3 + 2x + x^2 - 2
To find critical points, we set f'(x) = 0 and solve for x:
-3x^3 + 2x + x^2 - 2 = 0
This equation is not easy to solve algebraically, so we can use numerical methods like Newton's method or graphing technology to find approximate values for the critical points.
Next, we need to classify the extrema as maximum or minimum. To do this, we use the second derivative test. We find the second derivative of f(x):
f''(x) = -9x^2 + 2 + 2x
Evaluate the second derivative at each critical point. If f''(x) > 0, then the point is a local minimum. If f''(x) < 0, then the point is a local maximum. If f''(x) = 0, the test is inconclusive.
Finally, to determine the intervals on which the function is increasing or decreasing, we examine the sign of the derivative f'(x). If f'(x) > 0, then the function is increasing. If f'(x) < 0, then the function is decreasing. The critical points divide the number line into intervals, and we can test the sign of f'(x) within each interval to determine whether it is increasing or decreasing.
For the function f(x) = 2x - x^3, to determine the end behavior, we look at the highest power of x in the function, which is x^3.
Since the coefficient of x^3 is negative (-1), the end behavior of the function is different for positive and negative values of x.
For positive values of x, as x approaches positive infinity, x^3 becomes large and negative, so -x^3 approaches negative infinity. Therefore, the function f(x) = 2x - x^3 approaches positive infinity as x approaches positive infinity.
For negative values of x, as x approaches negative infinity, x^3 becomes large and negative, so -x^3 approaches positive infinity. Therefore, the function f(x) = 2x - x^3 approaches negative infinity as x approaches negative infinity.