Question

Identify the open intervals on which the function is increasing or decreasing. (Enter your answers using interval notation.) y = x 16 − x2 increasing decreasing

Answer 1

To find where the function y = x±⁶ - x² is increasing or decreasing, we calculate the derivative y' = 16x±⁵ - 2x and find critical points. By testing intervals, we determine the function is increasing on (-∞, 0) and ((1/8), ∞), and decreasing on (0, (1/8)).

Step-by-step explanation:

Identifying Intervals of Increase and Decrease

To identify the open intervals on which the function y = x¹⁶ − x² is increasing or decreasing, we first need to find its derivative to determine the slope at any given point. We'll denote the derivative as y'. For the function y = x¹⁶ − x², the derivative is y' = 16x¹⁵ − 2x. Setting y' equal to zero and solving for x gives us the critical points where the function could change its behavior from increasing to decreasing or vice versa.

Setting y' to zero, we get 0 = 16x±⁵ − 2x. Factoring out x, we have x(16x±⁴ − 2) = 0, which gives us x = 0 and x = 1/8 as critical points where the slope is zero. We then test intervals around these points to determine where the function is increasing or decreasing.

For x < 0, y' > 0, so the function is increasing. For 0 < x < 1/8, y' < 0, so the function is decreasing, and for x > 1/8, y' > 0, so the function is increasing again. Therefore, the open intervals where the function is increasing are (-∞, 0) and ((1/8), ∞), and the open interval where the function is decreasing is (0, (1/8)).

Answer 2

The function y = x^16 - x^2 is increasing on the interval [0, +∞) since the first derivative is non-negative for x ≥ 0. No intervals of decrease exist for this function when x is non-negative.

Step-by-step explanation:

To determine the intervals on which the function y = x16 − x2 is increasing or decreasing, we first need to find the first derivative of the function, which gives us the slope of the tangent line at any given point on the graph. The first derivative, or the instantaneous rate of change of the function, is given by the formula:

f'(x) = 16x15 − 2x.

We then find the critical points by setting the first derivative equal to zero:

0 = 16x15 − 2x.

Solving for x, we find that x = 0 is a critical point. To determine where the function is increasing or decreasing, we look at the sign of the first derivative on either side of this critical point. Because the power of 16 in the first term is even and the coefficient is positive, it will always be positive or zero. The second term is always non-positive when x is non-negative. Thus, the first derivative will always be non-negative for x ≥ 0, which implies that the function is always increasing or constant (never decreasing).

In interval notation, the function y = x16 − x2 is increasing on the interval [0, +∞).