sketch a graph of the function f that is continuous on (-∞,∞) and has the following properties. f(x)<0…
sketch a graph of the function f that is continuous on (-∞,∞) and has the following properties. f(x)<0, f(x)>0 choose the correct graph below. o a. o b. o c.
Answer
Explanation:
Step1: Analyze the first - derivative
Since (f^{\prime}(x)<0), the function (f(x)) is decreasing for all (x\in(-\infty,\infty)). This means that as (x) increases, (y = f(x)) decreases. So we can rule out any graph that is increasing.
Step2: Analyze the second - derivative
Since (f^{\prime\prime}(x)>0), the function (f(x)) is concave up for all (x\in(-\infty,\infty)). A concave - up function has a graph that curves upward like a cup.
Step3: Check the options
Option A is increasing in some intervals, so it is not correct. Option B is increasing, so it is not correct. Option C is decreasing and concave up, which satisfies (f^{\prime}(x)<0) and (f^{\prime\prime}(x)>0).
Answer:
C.