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

9 months ago

Answered by James, an A Level Maths tutor with MyTutor

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


PremiumRebecca V. A Level Maths tutor, 13 plus  Maths tutor, GCSE Maths tuto...
View profile
£20 /hr

Rebecca V.

Degree: MA Logic and Philosophy of Mathematics (Masters) - Bristol University

Subjects offered: Maths, Physics+ 1 more

Further Mathematics

“Hi, I'm Rebecca, I'm 23 and I'm currently taking a year to myself just to study, doing my masters in Logic and Philosophy of Maths. To me, mathematics has always been black and white, right or wrong whilst philosophy is all about thin...”

Laura B. A Level Chemistry tutor, A Level Maths tutor, A Level Russia...
View profile
£20 /hr

Laura B.

Degree: Chemistry (Bachelors) - York University

Subjects offered: Maths, Russian+ 2 more

-Personal Statements-

“Hello! My name is Laura and I am very passionate about science! I am currently studying Chemistry with Biological and Medicinal Chemistry at University of York and I really hope that my passion about science will inspire you!  I am ve...”

Oliver W. A Level Chemistry tutor, GCSE Chemistry tutor, A Level Math...
View profile
£22 /hr

Oliver W.

Degree: Physical Natural Sciences (Chemistry) (Masters) - Cambridge University

Subjects offered: Maths, Chemistry


“Second year chemist, from the University of Cambridge, with several years experience tutoring in Chemistry and Core Mathematics.”

About the author

James F. A Level Maths tutor
View profile

James F.

Currently unavailable: for regular students

Degree: Economics (Bachelors) - Nottingham University

Subjects offered: 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

A rollercoaster stops at a point with GPE of 10kJ and then travels down a frictionless slope reaching a speed of 10 m/s at ground level. After this, what length of horizontal track (friction coefficient = 0.5) is needed to bring the rollercoaster to rest?

How do you integrate the function cos^2(x)

What is the best way to prove trig identities?

What's the deal with Integration by Parts?

View A Level Maths tutors


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