how to tell on what interval a function is increasing

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On what intervals does f(x) = (1/3)10³ + 2.5x²– 14x + 25 increase?

Possible Answers:

(–∞, –7)

(–∞, –seven), (–7, 2), and (2, ∞)

(–7, ii), and  (ii, ∞)

(2, ∞)

(–∞, –7) and (2, ∞)

Correct answer:

(–∞, –7) and (ii, ∞)

Explanation:

We volition use the tangent line slope to ascertain the increasing / decreasing of f(x). To this end, allow us begin past taking the first derivative of f(x):

f'(x) = ten^two + 5x – xiv

Solve for the potential relative maxima and minima by setting f'(x) to 0 and solving:

x^two + 5x – 14 = 0; (ten – 2)(x + 7) = 0

Potential relative maxima / minima: x = 2, x = –7

We must test the post-obit intervals: (–∞, –vii), (–7, 2), (ii, ∞)

f'(–10) = 100 – fifty – 14 = 36

f'(0) = –14

f'(10) = 100 + 50 – xiv = 136

Therefore, the equation increases on (–∞, –vii) and (2, ∞)

Find the interval(s) where the following part is increasing. Graph to double check your answer.

Possible Answers:

Always

Never

Correct respond:

Explanation:

To discover when a function is increasing, you must first take the derivative, then set it equal to 0, and then find between which cypher values the function is positive.

Commencement, take the derivative:

Set equal to 0 and solve:

Now test values on all sides of these to observe when the office is positive, and therefore increasing. I volition examination the values of -6, 0, and 2.

Since the values that are positive is when x=-6 and 2, the interval is increasing on the intervals that include these values. Therefore, our respond is:

Find the interval(s) where the post-obit function is increasing. Graph to double check your answer.

Possible Answers:

Always

Never

Right answer:

Explanation:

To find when a office is increasing, you must first take the derivative, and so set it equal to 0, and then find betwixt which nil values the function is positive.

First, accept the derivative:

Set up equal to 0 and solve:

Now test values on all sides of these to discover when the function is positive, and therefore increasing. I will test the values of 0, 2, and ten.

Since the value that is positive is when x=0 and 10, the interval is increasing in both of those intervals. Therefore, our respond is:

Is  increasing or decreasing on the interval ?

Possible Answers:

Increasing. on the interval.

Increasing. on the interval.

Cannot exist determined from the information provided

Decreasing. on the interval.

Decreasing. on the interval.

Correct answer:

Increasing. on the interval.

Explanation:

To find increasing and decreasing intervals, we need to find where our first derivative is greater than or less than zero. If our first derivative is positive, our original function is increasing and if g'(10) is negative, g(x) is decreasing.

Begin with:

If nosotros plug in any number from three to vi, we get a positve number for chiliad'(x), So, this function must be increasing on the interval {iii,6}, because k'(x) is positive.

Is  increasing or decreasing on the interval ?

Possible Answers:

Increasing, because  is positive.

Decreasing, because  is positive.

Decreasing, because  is negative.

Increasing, because is negative.

 is neither increasing nor decreasing on the given interval.

Correct answer:

Increasing, considering  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.

And so starting with:

We get:

 using the Power Rule .

Notice the function on each end of the interval.

So the offset derivative is positive on the whole interval, thus g(t) is increasing on the interval.

Is the following function increasing or decreasing on the interval ?

Possible Answers:

Decreasing, because  is positive on the given interval.

The part is neither increasing nor decreasing on the interval.

Increasing, considering  is positive on the given interval.

Decreasing, because  is negative on the given interval.

Increasing, considering  is negative on the given interval.

Correct answer:

Increasing, because  is positive on the given interval.

Explanation:

A part is increasing on an interval if for every bespeak on that interval the first derivative is positive.

Then we need to find the start derivative so plug in the endpoints of our interval.

Observe the first derivative past using the Power Rule

Plug in the endpoints and evaluate the function.

Both are positive, and then our function is increasing on the given interval.

On which intervals is the following function increasing?

Correct respond:

Caption:

The first step is to observe the get-go derivative.

Recollect that the derivative of

Side by side, find the critical points, which are the points where  or undefined. To find the  points, set the numerator to , to find the undefined points, gear up the denomintor to . The critical points are and

The final step is to endeavor points in all the regions  to see which range gives a positive value for .

If we plugin in a number from the kickoff range, i.e , nosotros get a negative number.

From the second range,, we become a positive number.

From the third range, , we get a negative number.

From the last range, , we get a positive number.

So the 2d and the terminal ranges are the ones where  is increasing.

Beneath is the complete graph of . On what interval(s) is increasing?

Correct answer:

Explanation:

 is increasing when is positive (to a higher place the -axis). This occurs on the intervals.

Function A

Part B

Function C

Office D

Role E

v graphs of different functions are shown above. Which graph shows anincreasing/non-decreasing office?

Possible Answers:

Office D

Function E

Part B

Function A

Function C

Right answer:

Office E

Explanation:

A function is increasing if, for whatsoever, (i.eastward the slope is ever greater than or equal to nothing)

Function E is the merely function that has this belongings. Annotation that function E is increasing, but notstrictly increasing

Find the increasing intervals of the following function on the interval :

Right answer:

Explanation:

To find the increasing intervals of a given function, one must decide the intervals where the office has a positivefirstderivative. To find these intervals, get-go discover the disquisitional values, or the points at which the first derivative of the function is equal to aught.

For the given function,.

This derivative was found by using the power rule

.

When fix equal to zero,. Because we are just considering the open interval (0,five) for this function, we can ignore. Adjacent, nosotros look the intervals around the critical value, which are and . On the first interval, the first derivative of the function is negative (plugging in values gives us a negative number), which ways that the role is decreasing on this interval. However for the 2nd interval, the first derivative is positive, which indicates that the function is increasing on this interval.

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