Over which interval is the graph of f(x) = x2 + 5x + 6 increasing?

Asked by maham237 @ 25/12/2020 in Mathematics viewed by 346 People

Over which interval is the graph of f(x) = x2 + 5x + 6 increasing? (–6.5, ∞) (–5, ∞) (–∞, –5) (–∞, –6.5)

Answered by maham237 @ 25/12/2020

To check over which interval the graph of our function is increasing, we are going to use the derivative test. The derivative test uses the derivative of the function to locate its critical points, and then check over which intervals the derivative of the function is positive or negative. If the derivative of the function is positive over an interval, then the function is increasing over that interval, but if the derivative of the function is negative over an interval, then the function is decreasing over that interval.

The first thing we are going to do is take the derivative of our function

$f(x)=x^2+5x+6$

$f'(x)=\frac{dy}{dx} x^2+\frac{dy}{dx}5x+\frac{dy}{dx}6$

$f'(x)=2x+5$

Next, we are going to set the derivative equal to zero and solve for $x$ to find the critical points of our function

$2x+5=0$

$2x=-5$

$x=-\frac{5}{2}$

Our function only has only critical point $x=-\frac{5}{2}$ . That critical point divides the domain of our function in tow intervals: (-∞,-2.5) and (-2.5,∞)

Now we are going to take test values in each interval and evaluate the derivative for those test values

- For the interval (-∞,-2.5), we are going to use the test value -2.6

$f'(x)=2x+5$

$f'(-2.6)=2(-2.6)+5$

$f'(-2.6)=-5.2+5$

$f'(-2.6)=--0.2$

Since the derivative is negative over interval (-∞,-2.5), the function $f(x)=x^2+5x+6$ is decreasing over the interval (-∞,-2.5).

- For the interval (-2.5,∞), we are going to use the test value -2.4

$f'(x)=2x+5$

$f'(-2.4)=2(-2.4)+5$

$f'(-2.4)=-4.8+5$

$f'(-2.4)=0.2$

Since the derivative is positive over interval (-2.5,∞), the function $f(x)=x^2+5x+6$ is increasing over the interval (-2.5,∞).

We can conclude that the graph of f(x) = x2 + 5x + 6 is increasing over the interval (-2.5,∞); therefore none of your given choices is the correct answer.

Over which interval is the graph of f(x) = x2 + 5x + 6 increasing?

Similar Questions

What are the zeroes of f(x) = x2 + 5x + 6?

Over which interval is the graph of f(x) = –x2 + 3x + 8 increasing?

Over what interval is the graph of f(x) = –(x + 8)2 – 1 decreasing?

Which is the graph of f(x) = x2 – 2x + 3?

Which of the following graphs represents the function f(x) = x2 + x − 6?

Which graph has the same end behavior as the graph of f(x) = –3x3 – x2 + 1?

Which is the graph of the function f(x) = x2 + 2x – 6?

The graph of f(x) = x6 – 2x4 – 5x2 + 6 is shown below.

Solve for x x2 5x a2 a 6 0

Which graph has a negative rate of change for the interval 0 to 2 on the x-axis?

Which of the following graphs could be the graph of the function f(x) = x4 + x3 – x2 – x?

What is the completely factored form of f(x) = x3 – 2x2 – 5x + 6?

The graph of f(x) = x2 is translated to form g(x) = (x – 2)2 – 3.

The graph of f(x) = x2 is translated to form g(x) = (x – 5)2 + 1.

What are the zeroes of f(x) = x2 − 6x + 8?

Most viewed questions in Mathematics

Unit 6 similar triangles homework 2 similar figures answer key

All things algebra unit 7 polygons and quadrilaterals answer key

Gina wilson all things algebra 2016 answer key unit 11

Unit 1 geometry basics homework 4 angle addition postulate answers

Unit 6 exponents and exponential functions homework 8 answer key

The manager of a vacation resort believes that the ages

What is 271 403 rounded to the nearest ten thousand

Unit 1 geometry basics homework 3 angle relationships answer key

Homework 2 parallelograms

Why would a prism beat a sphere in a competition?

John wants to send a letter to peter

An artist sells a painting for 20.50 but uses 5.75

Unit 4 congruent triangles homework 1 classifying triangles answer key

Unit 4 congruent triangles homework 2 angles of triangles

Compare the values of the 2s and 5s in 55220