Concavity..? (1 Viewer)

[Finx]

So, there's always a part in a question which asks "Find for which values of x is f(x) concave up [or down]" - usually for some cubic curve for Geometrical Applications of Differentiation.

I'm not 100% sure how to approach this. I know there's a way to do it graphically, but I'd like to know the algebraic method please.

Thanks in advance!

 

[boxhunter91]

ahh yes to do this you need an expression for f"(x) say f"(x)= x-1
If it is maximum then f"(x)<0
.: x-1<0
.:x<1

 

[untouchablecuz]

So, there's always a part in a question which asks "Find for which values of x is f(x) concave up [or down]" - usually for some cubic curve for Geometrical Applications of Differentiation.

I'm not 100% sure how to approach this. I know there's a way to do it graphically, but I'd like to know the algebraic method please.

Thanks in advance!

well

if y=f(x) is concave down at x=a, then f''(a)<0 => solve f''(x)<0 to find the range of x values for which f(x) is concave down

if y=f(x) is concave up at x=a, then f''(a)>0 => solve f''(x)>0 to find the range of x values for which f(x) is concave up

e.g.

f(x)=ax^3+bx^2+cx+d
f'(x)=3ax^2+2bx+c
f''(x)=6ax+2b

for points at which f(x) is concave down,

f''(x)<0 => 6ax+2b<0 and hence x<-b/3a

 

[addikaye03]

Concave up- f''(x)>0

Concave down- f''(x)<0

 

[Darnie]

A curve will be concave up when f''(x) > 0

It will be concave down when f''(x) < 0

 

[Finx]

But you use f"(x) > 0 and f"(x) < 0 for testing the nature of stationary points, don't you?

 

[kelllly]

Another example for you :

Let's say the question is 'find for which values of x is f(x) = x^3 + x^2 concave down.'

1. First, find the first and second derivative:

f'(x) = 3x^2 + 2x
f''(x) = 6x + 2

2. Then, find the values of x for which the second derviative is negative*.

f''(x) < 0
6x + 2 < 0
6x < -2
x < -1/3

Answer: For x < -1/3 is f(x) = x^3 + x^2 concave down.


*I remember it like this:

Concave down = Unhappy = Negative
f''(x) < 0

Concave up = Happy = Positive
f''(x) > 0

 

[kelllly]

But you use f"(x) > 0 and f"(x) < 0 for testing the nature of stationary points, don't you?

Yes, you do.

The second derivative is used to determine concavity.

Taking the example of a parabola:

A maximum stationary point corresponds with a concave down parabola.
A minimum stationary point corresponds with a concave up parabola.

 

[Finx]

Ah, thanks!

I'd been very confused, because the second derivative is used to both determine the nature of stationary points AND finding what x values give a concave up/down curve.

It's clear now. Thanks everyone!

 

[cutemouse]

*I remember it like this:

Concave down = Unhappy = Negative
f''(x) < 0

Concave up = Happy = Positive
f''(x) > 0

That 'mnemonic' seems popular amongst females.

 

[Makro]

Or smiley face and upside down smiley face for the General kids.

f'(x) = gradient
f''(x) = concavity

Two things that I just think are nice to know without having to think.

 

[Cloesd]

Or smiley face and upside down smiley face for the General kids.

f'(x) = gradient
f''(x) = concavity

Two things that I just think are nice to know without having to think.

What if the question asks, FIND ALL X for which a line is concave up. You go through the normal process yes..


f''(x) < 0
f''(x) = 12x(x-3)
12x(x-3) < 0

x < 3 and x < 0?

But if x is less than three, its already less than 0!, The back of the book says that its meant to be x is greater than 0, but less than three.

Or am i messing up the signs?

12x < 0
thus...
x > 0?

But that makes no sense.

What am i failing at?

 

[Lukybear]

Draw the parabola, hence x intercepts are 0 and 3. Shade below the graph to indicate <0.

I.e.
\___/
0 \..../ 3

thus 0<x<3

 

[kelllly]

What if the question asks, FIND ALL X for which a line is concave up. You go through the normal process yes..


f''(x) < 0
f''(x) = 12x(x-3)
12x(x-3) < 0

12x(x - 3) > 0
Concave up means f''(x) > 0

As Lukybear said, draw the parabola (attached).

From the graph, 12x(x - 3) > 0 (or positive) when:
x < 0
x > 3


BUT since you say the answer is 0 < x < 3, I assume you meant 'concave down'?

In that case, from the graph (again), 12x(x - 3) < 0 (or negative) when:
0 < x < 3

 

[Albertingu]

f'"(x) give cocavtiy

 

[kwabon]

Concave up is a cup,
Concave down is a frown.


i should be a poet.