if f(x)=e^x g(x), where g(0) and g'(0)=5, find f'(0) and f(2)=10 and f'(x)=x^2 f(x) for all x, find f''(2)

Question

if f(x)=e^x g(x), where g(0) and g'(0)=5, find f'(0)

 

and 

 

f(2)=10 and f'(x)=x^2 f(x) for all x, find f''(2)

Answer 1

Find f'(x) from f(x).

 

f'(x) = e^xg(x) + e^xg'(x)

 

f'(0) = e⁰g(0) + e⁰g'(0)

 

f'(0) = 5 + 5

 

f'(0) = 10

 

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f''(x) = 2xf(x) + x²f'(x)

 

f"(2) = 2(2)f(2) + 2²x²

 

f"(2) = 40 + 4x²