Question

Let f(x) be a continuous function which satisfies the following conditions. f(0)=4;f(2)=2;f(5)=6

f ′ (0)=f ′ (2)=0
f ′ (x)>0 if ∣x−1∣>1
f ′ (x)<0 if ∣x−1∣<1
f ′′ (x)<0 if x<1 or if ∣x−4∣<1
f '′ (x)>0 if ∣x−2∣<1 or if x>5
​Find the intervals on which f is increasing. (Enter your answer as a comma-separated list of intervals.) (b) Find the intervals on which f is decreasing. (Enter your answer as a comma-soparated list of intervals.) (c) Find the largest open intervals on which f is concave up. (Enter your answer as a comma-separated list of intervals.) स (d) Find the largest open intervals on which f is concave down. (Enter your answer as a comma-separated list of intervals.)

Answer 1

To find the intervals on which f is increasing, we look at the sign of the derivative f'(x). When f'(x) is positive, the function is increasing. To find the intervals on which f is decreasing, we look at the sign of f'(x) when |x-1| < 1. To find the intervals on which f is concave up, we look at the sign of the second derivative f''(x). And to find the intervals on which f is concave down, we look at the sign of f''(x) when x < 1 or |x-4| < 1.

Step-by-step explanation:

To find the intervals on which f is increasing, we need to look at the sign of the derivative f'(x). When f'(x) is positive, the function is increasing. We know that f'(x) is positive when |x-1| > 1. Therefore, the function is increasing on the intervals (-∞,0) and (2,5).

To find the intervals on which f is decreasing, we look at the sign of f'(x) when |x-1| < 1. Since f'(x) is negative in those cases, the function is decreasing on the intervals (0,2) and (5,∞).

To find the largest open intervals on which f is concave up, we look at the sign of the second derivative f''(x). We know that f''(x) is positive when |x-2| < 1 or when x > 5. Therefore, the function is concave up on the intervals (-∞,2) and (5,∞).

To find the largest open intervals on which f is concave down, we look at the sign of f''(x) when x < 1 or |x-4| < 1. Since f''(x) is negative in those cases, the function is concave down on the intervals (1,4).