Rolle's Theorem

[Jason76]

Verify Rolle's Theorem for the interval. Find what \(\displaystyle x\) value satisfies it.

\(\displaystyle x^{3} - x^{2} - 2x + 9\) for interval \(\displaystyle [0,2]\)

1st check: This is a polynomial so differentiable and continuous

\(\displaystyle f(0) = (0)^{3} - (0)^{2} - 2(0)+ 9 = 9\)

\(\displaystyle f(2) = (2)^{3} - (2)^{2} - 2(2)+ 9 = 9\)

2nd check: \(\displaystyle f(0) = f(2)\)

\(\displaystyle f'(x) = 3x^{2} - 2x - 2 \)

\(\displaystyle 3x^{2} - 2x - 2 = 0\)

What two numbers have a product of \(\displaystyle -6\) and a sum of \(\displaystyle -2\) ?

\(\displaystyle (?)(?) = 0\)

 

[stapel]

\(\displaystyle 3x^{2} - 2x - 2 = 0\)

What two numbers have a product of \(\displaystyle -6\) and a sum of \(\displaystyle -2\) ?

If you'd taken algebra, you would have encountered something called "the Quadratic Formula", which is very helpful for solving the above equation.

 

[Deleted member 4993]

Verify Rolle's Theorem for the interval. Find what \(\displaystyle x\) value satisfies it.

\(\displaystyle x^{3} - x^{2} - 2x + 9\) for interval \(\displaystyle [0,2]\)

1st check: This is a polynomial so differentiable and continuous

\(\displaystyle f(0) = (0)^{3} - (0)^{2} - 2(0)+ 9 = 9\)

\(\displaystyle f(2) = (2)^{3} - (2)^{2} - 2(2)+ 9 = 9\)

2nd check: \(\displaystyle f(0) = f(2)\)

\(\displaystyle f'(x) = 3x^{2} - 2x - 2 \)

\(\displaystyle 3x^{2} - 2x - 2 = 0\)

What two numbers have a product of \(\displaystyle -6\) and a sum of \(\displaystyle -2\) ?

\(\displaystyle (?)(?) = 0\)

You have been shown the method in http://www.freemathhelp.com/forum/threads/84095-Factoring-Problem-with-12

 

[HallsofIvy]

Verify Rolle's Theorem for the interval. Find what \(\displaystyle x\) value satisfies it.

\(\displaystyle x^{3} - x^{2} - 2x + 9\) for interval \(\displaystyle [0,2]\)

1st check: This is a polynomial so differentiable and continuous

\(\displaystyle f(0) = (0)^{3} - (0)^{2} - 2(0)+ 9 = 9\)

\(\displaystyle f(2) = (2)^{3} - (2)^{2} - 2(2)+ 9 = 9\)

2nd check: \(\displaystyle f(0) = f(2)\)

\(\displaystyle f'(x) = 3x^{2} - 2x - 2 \)

\(\displaystyle 3x^{2} - 2x - 2 = 0\)

What two numbers have a product of \(\displaystyle -6\) and a sum of \(\displaystyle -2\) ?

\(\displaystyle (?)(?) = 0\)

The two numbers are \(\displaystyle 1+\sqrt{7}\) and \(\displaystyle 1-\sqrt{7}\). Why are you asking that question? If you are trying to factor that to try to solve the equation, especially since you are doing Calculus problems, you certainly should know that only a very few quadratic polynomials can be factored with integer coefficients. To solve such an equation, complete the square or use the quadratic formula.

 

[Jason76]

The two numbers are \(\displaystyle 1+\sqrt{7}\) and \(\displaystyle 1-\sqrt{7}\). Why are you asking that question? If you are trying to factor that to try to solve the equation, especially since you are doing Calculus problems, you certainly should know that only a very few quadratic polynomials can be factored with integer coefficients. To solve such an equation, complete the square or use the quadratic formula.

Ok, I will try it.