Verify that the function satisfies the three hypotheses of Rolle?s Theorem on th
ID: 2850050 • Letter: V
Question
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. (Enter your answers as a comma-separated list.) f(x)=x^3-x^2-12x+5, [0,4] Please try again. The three hypotheses of Rolle?s Theorem for a function for the given interval [a, b) are: (i) f is continuous on [a, b), (ii) f is differentiable on (a, b), and (iii) f(a) = f(b). Verify each of these, and then use the quadratic formula to solve for c in f(c) = 0 to obtain all numbers c in (a, b) that satisfy the conclusion of Rolle?s Theorem. 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. (Enter your answers as a comma-separated list.)Explanation / Answer
Solution:
1)f(x)=x3 -x2-12x+5 [0,4]
f(0)=5
f(4)=(4)3 -(4)2-12(4)+5=64-16-48+5=5
f(4)=5
f(0)=5=f(4)
f is continuous on [0,4] and f is differntiable on(0,4)
to find c ;f'(c)=0
f'(c)=3x2-2x-12
(b±sqrt(b24(a)(c)))/2(a)=
(2±(2)24(3)(12))/2(3)
(2±sqrt(148))/6
=(2±sqrt(4×37))/6
=(2±2sqrt(37))/6
c=(1±1sqrt(37))/3
c=2.36or -1.69
2)f(x)=sqrt(x)-1/3x [0,9]
f(0)=0
f(9)=sqrt(9)-1/3(9)=0
f(0)=f(9)=0
f is continuous on [0,9] and f is differntiable on(0,9)
f'(c)=0
f'(x)=1/2sqrt(x)-1/3=0
1/2sqrt(x)=1/3
2sqrt(x)=3
4x=9
c=3/2,-3/2
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