Verify that the function satisfies the three hypotheses of Rolle’s Theorem on the given interval. Then find all numbers c that satisfy the conclusion of Rolle’s Theorem. f(x)=5−12x+3x2, [1,3]

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Verify that the function satisfies the three hypotheses of Rolle’s Theorem on the given interval. Then find all numbers c that satisfy the conclusion of Rolle’s Theorem. f(x)=5−12x+3x2, [1,3]

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Rolle&#039;s Theorem says that is a number c in (a,b) such that f′(c)=0 if f satisfies the following hypotheses:Because given function is a classical polynomial functions, the first two hypotheses are naturally satisfied. Now we&#039;ll check the third hypotheses:f(x)=5−12⋅x+3⋅x2→f(a)=f(1)=5−12⋅1+3⋅12=−4→f(b)=f(3)=5−12⋅33⋅32=−4Because f(1)=f(3) is satisfied, the third hypothesis is also satisfied. Hence Rolle&#039;s theorem is applicable on given interval. To find all numbers c that satisfy the conclusion of Rolle&#039;s theorem, we&#039;ll differentiate our function:f(x)=5−12⋅x+3⋅x2f′(x)=5′−(12⋅x)′+(3⋅x2)′=0−12⋅1+2⋅3⋅x=−12+6⋅xNow we&#039;ll find the value of c:f(c)=0−12+6⋅c=06⋅c=12c=2Result: c=2

Verify that the function satisfies the three hypotheses of Rolle’s Theorem on th

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