Sketching curves Sketch a graph of a function f that is contin...

Question

Sketching curves Sketch a graph of a function f that is continuous on (-∞,∞) and has the following properties.

f'(x) < 0 and f"(x) > 0 on (-∞,0); f'(x) > 0 and f"(x) < 0 on (0,∞)

Accepted Answer

Hi everyone, let's take a look at this practice problem. This problem says which of the following properties correctly describes the given graph. But the question we're given a graph of a function F of X, and this function is decreasing everywhere, and it has an inflection point at X equals 1. For X less than 1, our function is concave up, and for X greater than 1, our function is concave down. We're given 4 possible choices as our answers. For choice A, we have F of X is greater than 0 on the open interval from minus infinity to infinity. FX is greater than 0 on the open interval from minus infinity to 1, and F of X is less than 0 on the open interval from 1 to infinity. For choice B, we have F of X is greater than 0 on the open interval from minus infinity to infinity. F of X is less than 0 on the open interval from minus infinity to 1. FX is greater than 0 on the open interval from 1 to infinity. And for choice C, F of X is less than 0 on the open interval from minus infinity to infinity. F of X is less than 0 on the open interval from minus infinity to 1. FX is greater than 0 on the open interval from 1 to infinity. For choice D, F of X is less than 0 on the open interval from minus infinity to infinity. F of X is greater than 0 on the open interval from minus infinity to 1, and F of X is less than 0 on the open interval from 1 to infinity. Now, to answer this question, we need to determine the intervals over which our 1st and 2nd derivative are either positive or negative. So let's start by looking at the first derivative. Now, based upon the graph, we see that our function F of X is constantly decreasing, and it's decreasing everywhere. Recall that if a function is decreasing, that means its first derivative is negative. Therefore, F of X, It's going to be less than 0 on the open interval from minus infinity to infinity, so for all X. Now we need to look at the 2nd derivative. And recall that the second derivative is positive whenever our function is concave up, and it is um negative whenever the function is concave down. Now, our function has an inflection point at X equal to 1, and for X less than 1, our function is going to be concave up. And so therefore, our second derivative is going to be positive over that interval. Therefore, F of X is going to be greater than 0 on the open interval from minus infinity to 1. Now, if we look at the values of X greater than 1, we see that our function is concave down. And since our function is concave down, that means the second derivative is negative. Therefore, F of X is going to be less than 0 on the open interval from 1 to infinity. So, these three statements are the properties that we're looking for for the question. And if we compare our results here to our answer choices, we see that the correct answer choice corresponds to answer choice D. So I hope that this explanation has been useful, and I'll see you in the next video.

Sketching curves Sketch a graph of a function f that is continuous on (-∞,∞) and has the following properties.

f'(x) < 0 and f"(x) > 0 on (-∞,0); f'(x) > 0 and f"(x) < 0 on (0,∞)

Key Concepts

Derivative and Its Significance

Second Derivative and Concavity

Continuity of Functions

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