(x+2)^3 (x-3)^4 verify rolle's theorem for following function [-2,3]
Answers
Answer:
Rolle’s Theorem: If f:R→R is continuous on [a,b] , differentiable on (a,b) , and f(a)=f(b) , then ∃c∈(a,b) where f′(c)=0 .
First, verify that prerequisites are met.
PrerequisiteContinuous on [a,b]Differentiable on (a,b)f(a)=f(b)Completion✓✓✓
Our goal is to take the derivative and set it equal to 0. Then, solve it to find some number c and check that that number c is in the interval (−2,3) .
Step 1: Take the derivative.
ddx[f(x)]=f′(x)=ddx[(x+2)3(x−3)4]=ddx[(x+2)3](x−3)4+(x+2)3ddx[(x−3)4]=3(x+2)2(x−3)4+4(x+2)3(x−3)3=(x+2)2(x−3)3[3(x−3)+4(x+2)]=(x+2)2(x−3)3(7x−1)
Step 2: Set it equal to 0.
(x+2)2(x−3)3(7x−1)=0⎧⎩⎨⎪⎪⎪⎪(x+2)2=0⟹x=−2(x−3)3=0⟹x=37x−1=0⟹x=17
Step 3: Form conclusion.
As you can see, endpoints are included but not considered as we are looking for c∈(−2,3) . It’s rather trivial since f(a)=f(b) , in general, will always form a horizontal line with slope of 0 . So, our only valid c is c=17 which is in fact in [−2,3] , therefore Rolle’s Theorem applies.
