Q. 41 Determine whether or not each fu... [FREE SOLUTION]

Chapter 3: Q. 41 (page 249)

Determine whether or not each function f in Exercises 41鈥�48 satisfies the hypotheses of Rolle鈥檚 Theorem on the given interval [a, b]. For those that do, use derivatives and algebra to find the exact values of all c 鈭� (a, b) that satisfy the conclusion of Rolle鈥檚 Theorem.
$f (x) = x^{3} - 4 x^{2} + 3 x, [a, b] = [0, 3]$

Short Answer

Expert verified

The function $f (x) = x^{3} - 4 x^{2} + 3 x$ satisfies the hypotheses of Rolle's theorem. The exact values of c that satisfies conclusion of Rolle's theorem are $c = \frac{4 + \sqrt{7}}{3}$ and $c = \frac{4 - \sqrt{7}}{3}$ .

Step by step solution

Step 1. Given information.

Consider the given function $f (x) = x^{3} - 4 x^{2} + 3 x, [a, b] = [0, 3]$ .

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 $[0, 3]$ and differentiable on $(0, 3)$ .

Now, find $f (0)$ and $f (3)$ .

$\begin{array}{rcl} f (0) & = & 0 \\ f (3) & = & 27 - 36 + 9 \\ = & 0 \end{array}$

So, $f (0) = f (3)$ .

Thus, the Rolle's theorem applies on $[0, 3]$ then there must exists some value $c 鈭� (0, 3)$ such that $f' (c) = 0$ .

Step 3. Find the exact values of c.

The function is $f (c) = x^{3} - 4 x^{2} + 3 x$ .

Differentiate the function with respect to c.

$f' (c) = 3 c^{2} - 8 c + 3$

Solve $f' (c) = 0$ .

$3 c^{2} - 8 c + 3 = 0 c = \frac{4 + \sqrt{7}}{3}, c = \frac{4 + \sqrt{7}}{3}$

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Short Answer

Step by step solution

Step 1. Given information.

Step 2. Satisfy hypotheses of Rolle's theorem.

Step 3. Find the exact values of c.

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