MYTUTOR SUBJECT ANSWERS

2023 views

Find the coordinates of the minimum point of the curve y = 3x^(2) + 9x + 10

(Note: in this answer I will use the notation y' to represent the differential of y, and by the same reasoning y'' denotes the differential of y', or the second derivative of y.)

A curve's maximum or minimum will be found at the stationary point of the curve (for a continuous function), where the gradient of the curve is flat and equal to zero.

We can find the gradient of a curve by differentiation, where the first derivative is found by the rule of 'bring the power down and minus 1 from the power.' Formally, this can be written as if y = Axn, then the differential of y with respect to x is y' = nAxn-1. Consider the curve y = 3x2 + 9x + 10 to see how this works in practice, first including the equation's hidden powers for more clarity:

y = 3x+ 9x+ 10x0

y' = (2)(3)x2-1 + (1)(9)x1-1 + (0)(10)x0-1

= y' = 6x + 9, as x= 1, and the last term disappears as it is multiplied by zero, constants (terms with no 'x' included) will always drop out once differentiated.

As stated earlier, the maximum or minimum is found where the curve is stationary, and the gradient is equal to zero, and so we set our newly derived gradient equal to zero, and rearrange the equation to find the value of x at this point:

6x + 9 = 0 (minus 9 from both sides)

6x = -9 (divide both sides by 6)

x = -9/6 = -3/2, and so the curve is stationary at the point where x is = to -3/2. What is the value of y at this point? Simply substitute this newly found x value into the original equation of the curve:

y = 3(-3/2)+ 9(-3/2) + 10 = (27/4) - (27/2) + 10 = 13/4. So there is a stationary point at (-3/2, 13/4).

Finally, in order to ensure that this point is a minimum point, we need to perform the second derivative test. This test states that we must differentiate y again (differentiate y'), if:

y'' > 0, then the curve is convex and minimised at that point.

y'' < 0, then the curve is concave and maximised at that point.

Therefore, lets differentiate y' = 6x + 9 again:

y'' = 6 which is > 0 and so the curve is in fact minimised, thus we can conclude that the minimum point of the curve is found at the point (-3/2, 13/4)

James F. A Level Maths tutor

2 years ago

Answered by James, an A Level Maths tutor with MyTutor


Still stuck? Get one-to-one help from a personally interviewed subject specialist

434 SUBJECT SPECIALISTS

£24 /hr

Sophie H.

Degree: Mathematics (Masters) - Bristol University

Subjects offered:Maths, Further Mathematics

Maths
Further Mathematics

“I’ll help anyone finding maths hard. I still struggle now, it's not meant to be easy! Be at my tutorial or (c^2-a^2) <- a maths joke, I apologise.”

Me V. A Level Maths tutor, GCSE Maths tutor, GCSE Physics tutor
£22 /hr

Vaikkun V.

Degree: Electrical and Electronic Engineering (Masters) - Imperial College London University

Subjects offered:Maths, Physics+ 2 more

Maths
Physics
Chemistry
.PAT.

“I am passionate, motivated, and always hungry for more knowledge. I can bring energy and life to the classroom, and engage the student.”

£26 /hr

Scott R.

Degree: PGCE Secondary Mathematics (Other) - Leeds University

Subjects offered:Maths, Further Mathematics

Maths
Further Mathematics

“I am currently completing 2 PGCEs in Leeds. I have always had a passion for maths and my objective is to help as many as possible reach their full potential.”

About the author

James F.

Currently unavailable: for regular students

Degree: Economics (Bachelors) - Nottingham University

Subjects offered:Maths

Maths

“Hi there everyone! My name is James and I'm a tutor for A-Level Maths. I have loved maths ever since I was young, but realise how daunting it can be for some, and would love to try to transfer some of my own methods and reasonings to ...”

MyTutor guarantee

You may also like...

Posts by James

Find the cartesian equation of a curve?

Find the coordinates of the minimum point of the curve y = 3x^(2) + 9x + 10

Other A Level Maths questions

How can I improve my mathematics

How do I find the coordinates of maximum and minimum turning points of a cubic equation?

Differentiate y = 4ln(x)x^2

I don't understand the point of differentiation or integration

View A Level Maths tutors

We use cookies to improve your site experience. By continuing to use this website, we'll assume that you're OK with this. Dismiss

mtw:mercury1:status:ok