Answer:
We seek to verify the Mean Value Theorem for the function
f(x)=3x2+2x+5
on the interval
[−1,1]
The Mean Value Theorem, tells us that if
f(x)
is differentiable on a interval
[a,b]
then ∃
c∈[a,b]
st:f'(c)=f(b)−f(a)b−a
So, Differentiating wrt
x
we have:
f'(x)=6x+2
And we seek a value
c∈[−1,1]
st: f'(c)=f(1)−f(−1)1−(−1)
∴6c+2=(3+2+5)−(3−2+5)2
∴6c+2=42
∴6c+2=2
∴6c=0
∴c=0
