Question

Which of the following intervals represents the interval of increase for the function f(x)=7+2x²-x⁴?

A) (−1,1)
B) (−2,2)
C) (−1,0) and (1,[infinity])
D) (−2,−1) and (0,2)

Answer 1

The intervals of increase for the function f(x) = 7 + 2x² - x⁴ are found by differentiating the function and solving for where the derivative is zero. The function is increasing on the intervals (-1,0) and (1,∞), which correspond to option (C).

Step-by-step explanation:

To find the intervals of increase for the function f(x) = 7 + 2x² - x⁴, we need to analyze the first derivative of the function, f′(x). The first derivative f′(x) gives us the slope of the tangent line to the curve at any given point, and when this is positive, the function is increasing. So let's differentiate:

f′(x) = 4x - 4x³

Now we find where f′(x) = 0 to determine potential intervals of increase:

0 = 4x - 4x³
0 = 4x(1 - x²)
0 = x(1 - x)(1 + x)

This gives us x = 0, x = 1, and x = -1. To determine the behavior of the function between these points, we can test values within each interval. Taking x = -0.5, 0.5, and 1.5, we find that the function is increasing on the intervals (-1, 0) and (0, 1). Thus, the correct answer is (C) (-1,0) and (1,∞).