# Fuction and Graph

1. Jan 22, 2007

### Harmony

1. The problem statement, all variables and given/known data
The function f is defined as
f(x)=3x^5-10x^3+1
1) Determine the maximum and minimum points, as well the point of inflection of the graph f.

2. Relevant equations
All the differentiation equation.

3. The attempt at a solution

I found the maximum point and minimum point, but I had some trouble with the inflection point.

So....
f''(x)=60x^3-60x^2
f''(x)=0,
60x^3-60x^2=0
x=0, x=1

f'''(0)=0

Hence the only inflection point is (1,-6)

2. Jan 22, 2007

### jing

Not sure this should be in this topic as it is calculus not precalculus.

Have another look at your differentiation, in particular check the term -60x^2 in your expression for f''(x).

3. Jan 22, 2007

### marlon

This second derivative is wrong. Calculate it again.

marlon

4. Jan 22, 2007

### HallsofIvy

Staff Emeritus
f"(x)= 60x3- 60x= 60x(x2- 1)= 0
for x= 0, 1, and -1.

One definition of "inflection point" is that the second derivative changes sign there. Yes, it is true that since f '''(0)= 0, the second derivative might NOT change signs there. For example, if f ''= x2, then while f''(0)= 0, f'' does not change signs there. But f'''(0)= 0 does not mean it CAN'T change signs there. For example, if f ''= x3, then, again, f'''(x)= 3x2 so f'''(0)= 0 but x3 does change signs there.

In this particular case, f''(1/2)= 60/8- 120< 0 while f''(-1/2)= -60/8+120> 0 so f'' does change signs at 0.