second derivitive problem help

[zzinfinity]

Here is the problem.
Suppose f(3)=2, f ' (3)= 5 and f "(3)= -2. Then d^2/dx^2 (f^2(x)) at x= 3 is equal to...

My first instinct would be simply to square f "(3) and get 4. However it is a multiple choice question and that is not one of the choices. How do you do this type of problem?

 

[mmm4444bot]

… (f^2(x)) …

I'm wondering what the above notation means.

 

[zzinfinity]

I was assuming it is the same as (f(x))^2. but I could be mistaken

 

[BigGlenntheHeavy]

\(\displaystyle Given: \ f(3) \ = \ 2, \ f'(3) \ = \ 5, \ and \ f"(3) \ = \ -2.\)

\(\displaystyle Find \ \frac{d^{2}[f^{2}(x)]}{dx^{2}} \ at \ x \ = \ 3.\)

\(\displaystyle \frac{d[f^{2}(x)]}{dx} \ = \ 2f(x)f'(x)\)

\(\displaystyle \frac{d^{2}[f^{2}(x)]}{dx^{2}} \ = \ \frac {d^{2}[2f(x)f'(x)]}{dx^{2}} \ = \ 2[f'(x)f'(x)+f(x)f"(x)]\)

\(\displaystyle = \ 2[f'(3)f'(3)+f(3)f"(3)] \ = \ 2[(5)(5)+(2)(-2)] \ = \ 42.\)