Question

Verify Roll's theorem for
f(x) = (x+2)² (x-3)⁴ in (-2,3)​

Answer 1

Step-by-step explanation:

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