Question

Find the slope of f (x)=(8-x^3)(sqrt2-x)

Answer 1

Answer:

Step-by-step explanation:

First we want the slope of the curve which is its derivative.

f(x) = 8*Sqrt(x) + 7

f'(x) = 4/Sqrt(x)

The Mean Value Theorem, says

f'(c) = (f(b) - f(a))/(b - a) b = 6, a = 3

f'(c) = (8*Sqrt(6) + 7 - (8*Sqrt(3) + 7))/(6 - 3)

= (8*Sqrt(6) + 7 - 8*Sqrt(3) - 7)/(3)

= (8*Sqrt(6) - 8*Sqrt(3))/(3)

f'(c)= (8/3)(Sqrt(6) - Sqrt(3))

f'(x) from a to b = f'(c)

4/Sqrt(x) = (8/3)(Sqrt(6) - Sqrt(3))

1/Sqrt(x) = (2/3)(Sqrt(6) - Sqrt(3))

1/x = (4/9)(Sqrt(6) - Sqrt(3))2

1/x = (4/9)(6 - 2(Sqrt(6)Sqrt(3)) +3)

1/x = (4/9)(9 - 2(Sqrt(6)Sqrt(3)))

1/x = 4 - (8/9)(Sqrt(6)Sqrt(3))

1/x = 4 - (8/3)(Sqrt(2))

x = 1/(4 - (8/3)(Sqrt(2)))

x = 4.3713 = c