Question

H(x)=5x³ −3x⁵. (a) Find the interval of increase.

A) (−[infinity],−1)
B) (−1,0)
C) (0,1)
D) (1,[infinity])

Answer 1

The function h(x) = 5x³ − 3xµ is increasing on the interval (1, +∞), as determined by calculating the derivative, setting it to zero to find critical points, and testing the sign of the derivative in intervals around these points. The correct option is D.

Step-by-step explanation:

To find the intervals of increase or decrease of the function h(x) = 5x³ − 3xµ, we need to follow these steps:

• First, calculate the derivative of the function, h'(x), which represents the slope of the tangent to the function at any point x. This will help us to identify where the function is increasing or decreasing.
• The derivative of h(x) is h'(x) = 15x² - 15x⁴.
• To find the critical points where the function might be changing from increasing to decreasing or vice versa, set the derivative equal to zero: 15x² - 15x⁴ = 0.
• Factor out the common terms: 15x²(1 - x²) = 0. This gives us the critical points x = 0 and x = ± 1.
• To determine the interval of increase, we examine the sign of the derivative h'(x) on the intervals determined by these critical points. Plotting these points on a number line helps.
• Test values in the intervals (-∞, -1), (-1, 0), (0, 1), and (1, +∞). You'll notice that h'(x) is positive in the intervals where h(x) is increasing.
• After testing, we find that the derivative h'(x) is positive (indicating an increasing function) on the interval (1, +∞), corresponding to option D.

Therefore, the function h(x) = 5x³ − 3xµ is increasing on the interval (1, +∞).