second derivitive problem help

zzinfinity

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Nov 12, 2009
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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?
 
Given: f(3) = 2, f(3) = 5, and f"(3) = 2.\displaystyle Given: \ f(3) \ = \ 2, \ f'(3) \ = \ 5, \ and \ f"(3) \ = \ -2.

Find d2[f2(x)]dx2 at x = 3.\displaystyle Find \ \frac{d^{2}[f^{2}(x)]}{dx^{2}} \ at \ x \ = \ 3.

d[f2(x)]dx = 2f(x)f(x)\displaystyle \frac{d[f^{2}(x)]}{dx} \ = \ 2f(x)f'(x)

d2[f2(x)]dx2 = d2[2f(x)f(x)]dx2 = 2[f(x)f(x)+f(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)]

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