Respuesta :
The numbers c that satisfies the conclusion of Rolle’s Theorem is 2.
Given that,
Function; [tex]f(x) = 5-12x+3x^{2} , [1, 3][/tex]
The function satisfies the three hypotheses of Rolle’s Theorem on the given interval.
We have to find,
The value which c satisfies the conclusion of Rolle’s Theorem?
According to the question,
Function; [tex]f(x) = 5-12x+3x^{2} , [1, 3][/tex]
Rolle's Theorem states that is a number c in (a, b) such that f′(c)=0 if f satisfies the following hypotheses:
The function is a classical polynomial function, the first two hypotheses are naturally satisfied. Now we'll check the third hypothesis:
[tex]f(x)=5-12\times x+3\times x^2\\\\f(a)=f(1)=5-12\times1+3\times 1^2= -7+3 =-4\\\\f(b)=f(3)=5-12\times3+3 \times 3^2= -31 + 27 = -4[/tex]
Here, [tex]\rm f(a) =f(1)=f(3)[/tex] is satisfied, the third hypothesis is also satisfied.
Rolle's theorem is applicable to a given interval. To find all numbers c that satisfies the conclusion of Rolle's theorem, we will differentiate our function:
If a function f is continuous on the closed interval and differentiable on the open interval then there includes a point x = c in (a, b) such that f ‘(c) =0.
Differentiate the given function,
[tex]f(x)=5-12x+3x^2\\\\f'(x)=0-12\times 1+2\times3 \times x\\\\f'(x) =-12+ 6x[/tex]
Therefore,
To find the value of c substitute f(x) = f(c) =0 in the equation,
[tex]-12+6 \times c=0\\\\6 \times c=12\\\\6c = 12\\\\c =\dfrac{12}{6}\\\\ c=2[/tex]
Hence, The numbers c that satisfies the conclusion of Rolle’s Theorem is 2.
For more details refer to the link.
https://brainly.com/question/2292493
