Find the quadratic polynomial ax^2+bx+c which best fits the function f(x)=8^x at x=0, in the sense that g(0)=f(0), and f'(0)=g'(0), and f''(0)=g''(0).
g(x)=?

Question

Find the quadratic polynomial ax^2+bx+c which best fits the function f(x)=8^x at x=0, in the sense that g(0)=f(0), and f'(0)=g'(0), and f''(0)=g''(0). 
g(x)=?

anonymous · Answer

First of all calculate the f(x), f'(x) and f''(x) values for the given function
$$f(x)=8^x$$$$f \prime(x)=8^x\ln8$$$$f \prime \prime(x)=8^x\ln^28$$Now we can find values for x=0$$f(0)=1$$$$f \prime(0)=\ln8$$$$f \prime \prime(0)=\ln^28$$Now consider the polynomial g(x) and it's derivatives$$g(x)=ax^2+bx+c$$$$g \prime(x)=2ax+b$$$$g \prime \prime(x)=2a$$We should also find these when x is 0$$g(0)=c$$$$g \prime(0)=b$$$$g \prime \prime(0)=2a$$As stated that f''(0)=g''(0) we can replace g''(0) with f''(0). So we get$$f \prime \prime(0)=2a$$$$\ln^28=2a$$$$a=(\ln^28)/2$$Repeat this process to determine the values for b and c$$b=\ln8$$$$c=1$$Now simply substitute these values into the original polynomial g(x) and you will get your desired function.

anonymous · Answer

Thanks for the problem, really made me think. Anyone feel free to correct me if I made a mistake. :)

anonymous · Answer

thank you for your help! it is correct i just checked