# Thread: Using Properties to determine derivative and original equation

1. ## Using Properties to determine derivative and original equation

A function f is defined for all real numbers and has the following properties.

(i) f(1)=5
(ii) f(3)=21
(iii) for all real values of a and b, f(a+b) - f(a)= kab+2b^2 where k is a fixed real number independent of a and b

a) Use a=1 and b=2 to find k.
b) Find f'(3)
c) Find f'(x) and f(x) for all real x

So I found out that k=-4, but I'm stuck on how to find the original equation and thus the derivative by the given info.

2. Originally Posted by ment2byours
A function f is defined for all real numbers and has the following properties.

(i) f(1)=5
(ii) f(3)=21
(iii) for all real values of a and b, f(a+b) - f(a)= kab+2b^2 where k is a fixed real number independent of a and b

a) Use a=1 and b=2 to find k.
b) Find f'(3)
c) Find f'(x) and f(x) for all real x

So I found out that k=-4, but I'm stuck on how to find the original equation and thus the derivative by the given info.
First of all, for (a), I think $\displaystyle k=4$, not $\displaystyle -4$. Here's why: $\displaystyle 21-5=k*1*2+2*2^2$. Thus $\displaystyle 16=2k+8$ and $\displaystyle k=4$.

Use the definition of the derivative (using a and b for x and h): $\displaystyle \lim_{b\to 0} \frac{f(a+b)-f(a)}{b}$

You know $\displaystyle f(a+b)-f(a)=4ab+2b^2$ so the above limit equals $\displaystyle \lim_{b\to 0} \frac{4ab+2b^2}{b}=4a$.

Thus $\displaystyle f'(a) = 4a$. Thus $\displaystyle f'(3)=12$.

3. Oh thanks!! I should have recognized that it was similar for the definition for a derivative. Sorry about getting k wrong, I tend to keep messing up my pos/neg signs. So the deriv of the original equation is f'(x)=4x?

4. Continuing with above solution

Now we have f'(x) = 4x

So $\displaystyle f(x) = 2x^2 + c$ where c is constant of integration

using f(1) = 5
5 = 2 + c

c = 3

so function is $\displaystyle f(x) = 2x^2 + 3$

5. No. $\displaystyle f'(x)=4x$. This makes $\displaystyle f(x)=2x^2+C$. Since $\displaystyle f(1)=5$, $\displaystyle f(x)=2x^2+3$.

EDIT: What arpitagarwal82 said.

6. YOU guys are awesome

7. No problem.

8. Originally Posted by ment2byours
YOU guys are awesome
You are welcome.
We are here to help.