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Chapter 3: Q. 41 (page 249)

Determine whether or not each function f in Exercises 41–48 satisfies the hypotheses of Rolle’s Theorem on the given interval [a, b]. For those that do, use derivatives and algebra to find the exact values of all c ∈ (a, b) that satisfy the conclusion of Rolle’s Theorem.

fx=x3-4x2+3x,a,b=0,3

Short Answer

Expert verified

The function fx=x3-4x2+3x satisfies the hypotheses of Rolle's theorem. The exact values of c that satisfies conclusion of Rolle's theorem are c=4+73and c=4-73.

Step by step solution

01

Step 1. Given information.

Consider the given function fx=x3-4x2+3x,a,b=0,3.

02

Step 2. Satisfy hypotheses of Rolle's theorem.

A polynomial function is continuous and differentiable everywhere. The given function is cubic polynomial. So, it is continuous on 0,3and differentiable on 0,3.

Now, find f0and f3.

f0=0f3=27-36+9=0

So, f0=f3.

Thus, the Rolle's theorem applies on 0,3then there must exists some valuec∈0,3such that f'c=0.

03

Step 3. Find the exact values of c.

The function is fc=x3-4x2+3x.

Differentiate the function with respect to c.

f'c=3c2-8c+3

Solve f'c=0.

3c2-8c+3=0c=4+73,c=4+73

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