Find the quadratic polynomial g(x)=ax2+bx+c which best fits the function f(x)=7x at x=0, in the sense that g(0)=f(0), and g'(0)=f'(0), and g''(0)=f''(0).
To find the quadratic polynomial g(x) that best fits the function f(x) = 7x at x = 0, we need to ensure that g(0) = f(0), g'(0) = f'(0), and g''(0) = f''(0).
Let's start by finding the values of f(0), f'(0), and f''(0):
f(x) = 7x
To find f(0), we substitute x = 0 into the function:
f(0) = 7(0) = 0
To find f'(0), we differentiate the function:
f'(x) = 7
f'(0) = 7
To find f''(0), we differentiate f'(x):
f''(x) = 0 (the first derivative of a constant is always zero)
f''(0) = 0
Now, let's write down the general form of the quadratic polynomial g(x) = ax^2 + bx + c.
g(x) = ax^2 + bx + c
Next, we substitute x = 0 into g(x) and equate it to f(0):
g(0) = a(0)^2 + b(0) + c = 0
c = 0 (since the other terms are multiplied by 0)
So, we have g(x) = ax^2 + bx.
Next, we differentiate g(x) with respect to x to find g'(x):
g'(x) = 2ax + b
Substituting x = 0 into g'(x) and equating it to f'(0):
g'(0) = 2a(0) + b = 7
b = 7
Now, we have g(x) = ax^2 + 7x.
Finally, we differentiate g'(x) with respect to x to find g''(x):
g''(x) = 2a
Substituting x = 0 into g''(x) and equating it to f''(0):
g''(0) = 2a = 0
a = 0
Therefore, the quadratic polynomial g(x) that best fits the function f(x) = 7x at x = 0, in the sense that g(0) = f(0), g'(0) = f'(0), and g''(0) = f''(0), is g(x) = 7x.