Verify that the function satisfies the three hypotheses of

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) = 1 − 24x + 4x²,    [2, 4]

Answer 1

f(a)= f(2)=1-48+16 = -21

f(b)=f(4)=1-96+64= -21

f(a)=f(b)

f(x) = 1-24x +4x^2

f'(x) =-24 +8x

f'(c) = -24 + 8c = 0

8c = 24

c = 3

f(x) is an upward opening parabola which is continuous and differentiable everywhere, including the interval [2,4]

f'(c)=f(3)=-24+8(3)=0

slope of the secant line from (2,-21) to (4,-21) = 0

the slope of f'(c) = f'(3) = 0 = the slope of the tangent line at (3,-35)= vertex =minimum point of the parabola

Answer 2

Answer 3

f(x) continuous on [2,4], differentiable on (2, 4) and f(2) = f(4) = - 21; so f'(c) = - 24 + 8x = 0; c = 3