Homework Statement
State the Mean Value Theorem and find a point which satisfies the conclusions of the Mean Value Theorem for f(x)=(x-1)3 on the interval [1,4].
2. The attempt at a solution
Mean Value Theorem:states that there exists a c∈(a,b) such that f'(c)=\frac{f(b)-f(a)}{b-a}...
But would my answer be acceptable in the exam? Or would I need something further? I think I understand the question, it's just I sometimes fail to get my ideas across on paper and that is what worries me more.
Can I just answer this question like this...
Limiting x to -∞, f(x) will approach -∞.
And as f(0)=1 and f'(x)>0 for (-∞,0) there exists only one root in the interval (-∞,0). [IVT]
As f(0)=1, f(4)=-31 and f'(x)<0 for (0,4) there exists only one root in the interval (0,4). [IVT]...
Homework Statement
Determine how many real numbers satisfy the equation x3-6x2+1=0. Give reasons for your answer naming any theorems you use.
2. The attempt at a solution
let:f(x)=x3-6x2+1
→f'(x)=3x2-12x
for stationary points:f'(x)=0
→3x(x-4)=0
∴x=0 or x=4
to determine...