Mean Theorem Value

Question

Let f(x) = x√(x+8). Answers the following questions.

1. Find the average slope ¯m of the function f on the interval [ − 8 , 0 ] .

Average slope: ¯ m =

2. Verify the Mean Value Theorem by finding a number c in (-8,0) such f’( c ) = ¯ m .

Answer: c =

for question 1, the slope I got 0 but it’s not the right answer.

Answer 1

f(x) = xsqr(x+8) for the interval [-8,0]

f(-8) = -8

f(0) = 0

slope of secant line from f(-8) to f(0) = -8/-8 = 1

It's continuous and differentiable

so there's a "c" where f'(c) = 1

f'(x) = derivative of xsqr(x+8) = x(1/2)/sqr(x+8) + sqr(x+8) = 1

x/2 + x+8 = sqr(x+8)

3x/2 +8 = sqr(x+8)

9x^2/4 +24x +64 = x+8

9x^2/4 +23x +56 = 0

9x^2 +92x + 224 = 0

x = c = -92/18 +/-(1/18)sqr(92^2 -36(224))