Question

Let f(x) be a continuous function which satisfies the following conditions. f(0)=4;f(4)=2;f(7)=6 f′(0)=f′(4)=0 f′(x)>0 if ∣x−2∣>2 f′(x)<0 if ∣x−2∣<2 f′′(x)<0 if x<3 or if ∣x−6∣<1 f′′(x)>0 if ∣x−4∣<1 or if x>7 Find the intervals on which f is increasing. (Enter your answer as a comma-separated list of intervals.)

Answer 1

Based on the given conditions about the first and second derivatives of function f, the intervals where f is increasing are (4, ∞).

Step-by-step explanation:

The intervals on which function f is increasing can be determined based on the given properties of f and its derivatives. The function f has a positive first derivative, f'(x), whenever \( |x-2| > 2 \). This inequality is satisfied when x < 0 or x > 4, but since the domain of f is not specified for x < 0, we only consider the interval x > 4.

The first derivative is zero at x = 0 and x = 4, indicating possible local minima or maxima. Additionally, based on the second derivative f''(x), f is concave down when x < 3 or within one unit of x = 6, and concave up within one unit of x = 4 or when x > 7. The first derivative is negative when \(|x-2| < 2\), which happens for 0 < x < 4. Hence, the function f is decreasing in this range.

Taking all this together, the function f is increasing on the intervals (4, ∞).