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This table is given:

Code:

```
x f(x) f'(x) g(x) g'(x)
1 6 4 2 5
2 9 2 3 1
3 10 -4 4 2
4 -1 3 6 7
```

*The functions f and g are differentiable for all real numbers, and g is strictly increasing. The table above gives the values of the functions and their first derivatives at selected values of x. The function h is given by h(x) = f(g(x)) - 6.*

a) Explain why there must be a value r for 1 < r < 3 such that h(r) = -5

b) Explain why there must be a value c for 1 < c < 3 such that h'(c) = -5

c) Let w be the function given by w(x) = integral of f(t)dt from 1 to g(x). Find the value of w'(3).

d) If g^-1 is the inverse function of g, write an equation for the line tangent to the graph of y = g^-1(x) at x=2

a) Explain why there must be a value r for 1 < r < 3 such that h(r) = -5

b) Explain why there must be a value c for 1 < c < 3 such that h'(c) = -5

c) Let w be the function given by w(x) = integral of f(t)dt from 1 to g(x). Find the value of w'(3).

d) If g^-1 is the inverse function of g, write an equation for the line tangent to the graph of y = g^-1(x) at x=2

So far, all I've established is that f and g are continuous, g'(x) will always be positive, and g(x) will always be less than g(x+1). Everything I know is just the obvious, I don't know where to go next! Can someone please give me the solution so I can quit worrying about this stupid problem?

Thanks for any help!