differentiable fuctions

Question

If f is differentiable in (0,6) & f'(4) =5 then find lim[ f(4) -f(x²)]/[2-x] where x approaches to 2

f'(x) = g(x) and g'(x) = -f(x) for all real x and f(5) =2 = f'(5) then find f²(10) + g²(10)

Answer 1

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Answer 2

Samagra,

 

This looks like two separate problems.

 

For this first one, when you plug in x = 2, you get 0/0, so you must use l'Hopital's Rule:

 

lim_x→2 [f(4) − f(x²)]/(2 − x) = lim_x→2 −2x f′(x²)/−2

 

                                         = lim_x→2 x f′(x²)

 

                                         = 2 · f′(4) = 10.

 

For the second one, what you are given is a system of differential equations which simplifies to a single second-order DE:

 

g″(x) = −g(x) or −f″(x) = −f′(x).

 

You also have the initial conditions f(5) = 2 and f′(5) = 2. You can then solve this using any appropriate method, then find g(x) as f′(x), and evaluate f²(10) + g²(10) directly.

 

If you need further explanation of the missing steps, I can supply them. It is better for you to finish the problem using just what I have shown you so far.