Subject optional. Email address: Your name:. Report an Error. Possible Answers:. Correct answer:. Explanation : To find when a function is increasing, you must first take the derivative, then set it equal to 0, and then find between which zero values the function is positive.
First, take the derivative: Set equal to 0 and solve: Now test values on all sides of these to find when the function is positive, and therefore increasing. Possible Answers: Cannot be determined from the information provided. Correct answer: Increasing. Explanation : To find increasing and decreasing intervals, we need to find where our first derivative is greater than or less than zero. Possible Answers: is neither increasing nor decreasing on the given interval.
Correct answer: Increasing, because is positive. Explanation : To find out if a function is increasing or decreasing, we need to find if the first derivative is positive or negative on the given interval. So starting with: We get: using the Power Rule. Find the function on each end of the interval. Possible Answers: Increasing, because is positive on the given interval.
The function is neither increasing nor decreasing on the interval. Correct answer: Increasing, because is positive on the given interval. Explanation : A function is increasing on an interval if for every point on that interval the first derivative is positive. So we need to find the first derivative and then plug in the endpoints of our interval. Find the first derivative by using the Power Rule Plug in the endpoints and evaluate the function.
Both are positive, so our function is increasing on the given interval. On which intervals is the following function increasing? Explanation : The first step is to find the first derivative. Remember that the derivative of Next, find the critical points, which are the points where or undefined. The critical points are and The final step is to try points in all the regions to see which range gives a positive value for. Explanation : is increasing when is positive above the -axis.
Possible Answers: Function E. Correct answer: Function E. Explanation : A function is increasing if, for any , i. Explanation : To find the increasing intervals of a given function, one must determine the intervals where the function has a positive first derivative. This derivative was found by using the power rule. Copyright Notice. View Calculus Tutors. Hatim Certified Tutor.
Kyujin Certified Tutor. William Certified Tutor. University of Toronto, Bachelor of Science, Criminology. For simplicity, take the position versus time graph. The derivative of the position with respect to time is the velocity.
So imagine that a ball is going up positive. That means the position during this interval is increasing becoming more positive. What is going on with the velocity curve? Well, for the ball to be increasing its position in the positive direction, it must have positive velocity.
Suddenly, the ball starts to turn around. At this instant, similarly to when you throw up a ball and it reaches its highest point, the derivative of the position curve is zero because there is no change in position with respect to time at that instance in time.
Then, the ball is falling down and its position, although positive , is decreasing. Thus, the velocity, derivative of the position w.
Naturally, as in the ball example, whenever the derivative goes from positive to zero to negative, the point at which the derivative is equal to zero would be a local maximum. If it is the other way around, negative to zero to positive, we would have a local minimum. So, x is the local maximum value. Hope this helpful.
Sign up to join this community. The best answers are voted up and rise to the top. Figure 3 shows examples of increasing and decreasing intervals on a function. Figure 3. While some functions are increasing or decreasing over their entire domain, many others are not. A value of the input where a function changes from increasing to decreasing as we go from left to right, that is, as the input variable increases is called a local maximum.
If a function has more than one, we say it has local maxima. Similarly, a value of the input where a function changes from decreasing to increasing as the input variable increases is called a local minimum. In this text, we will use the term local. Clearly, a function is neither increasing nor decreasing on an interval where it is constant.
A function is also neither increasing nor decreasing at extrema. To locate the local maxima and minima from a graph, we need to observe the graph to determine where the graph attains its highest and lowest points, respectively, within an open interval.
Like the summit of a roller coaster, the graph of a function is higher at a local maximum than at nearby points on both sides. The graph will also be lower at a local minimum than at neighboring points. Figure 5 illustrates these ideas for a local maximum. We see that the function is not constant on any interval.
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