Definition of Derivative to Find Constants A, B, and C

[panadaeyes]

Homework Statement

Ax^2 + Bx + C if neg. infinity < x </= 0
f(x) =
x^(3/2) cos (1/x) if 0 <x< pos. infinity

Use the definition of the derivative to determine all possible values of the constants A, B and C such that f'(0) exists. Cannot use differentiation formulas.

Homework Equations

definition of the derivative: lim h->0 of (f(a+h) -f(a))/h

The Attempt at a Solution

I used the definition of the derivative for Ax_2 + Bx + C and got an answer of B; but what am I supposed to be looking for? Would f'(0) just be B?
and do I have to do anything with the second part ( x^(3/2) cos 1/x)?

 

[mighty2000]

it is important to always thoroughly read and understand the problem before attempting to solve it. In this case, the problem is asking for all possible values of the constants A, B, and C that would make f'(0) exist. This means that we need to find values for A, B, and C that would make the derivative of the function f(x) exist at x=0.

To do this, we need to use the definition of the derivative, which is given in the problem. We need to take the limit as h approaches 0 of (f(0+h) - f(0))/h. In other words, we need to find the limit of the difference quotient as h approaches 0.

For the first part of the function, Ax^2 + Bx + C, we can use the definition of the derivative to find the limit. This will give us a value for B, since the limit will only exist if B exists. Therefore, B is the only constant that we need to determine for the first part of the function.

For the second part of the function, x^(3/2) cos (1/x), we need to be careful because the function is not defined at x=0. However, we are only interested in the limit as h approaches 0, which means we can ignore the fact that the function is not defined at x=0. We can use the definition of the derivative to find the limit, and this will give us a value for A. Therefore, A is the only constant that we need to determine for the second part of the function.

To summarize, we need to use the definition of the derivative to find values for A and B that would make the derivative of the function f(x) exist at x=0. We do not need to do anything with the second part of the function, as the limit will give us a value for A. So, to answer the question, B is the only constant that needs to be determined for f'(0) to exist.