Calculus Questions

Question

Verify that the function satisfies the three hypotheses of Rolle's Theorem on the given interval. Then find all numbers c that satisfies the conclusion of Rolle's Theorem. (Enter your answers as a comma-separated list.)

f(x) = 7 − 48x + 4x²,    [5, 7]

c =

Answer 1

f(x)=7-48x+4x^2

f'(x) = -48 +8x

f(5) = 7-48(5)+4(5)^2= 107-240=-133

f(7)= 7-48(7)+4(7^2)= 7+196-336= -133

secant line has slope=0

f'(c)=-48+8c=0

8c=48

c=48/8

c =6

f(x)= 7-48x+4x^2= 4(x^2-12x+36)+7-4(36)

=4(x-6)^2 -137

f(5)= -133

f(7)=-133

upward opening parabola with vertex (6, -137)

which is continuous and differentiable on the interval 5 to 7