If polynomial functions f(x) and g(x) satisfy d/dx {f(x)g(x)}=2x , f(0)=−3 , and g(0)=3,what is f(10)g(10) ?
To find f(10)g(10), we need to first find the functions f(x) and g(x) that satisfy the given conditions.
We know that d/dx {f(x)g(x)} = 2x. This means that if we differentiate the product of f(x) and g(x) with respect to x, we should get 2x.
Let's solve this differential equation:
d/dx {f(x)g(x)} = 2x
Using the product rule for differentiation, we can expand it as:
g(x) * df(x)/dx + f(x) * dg(x)/dx = 2x
Since we have f'(x) and g'(x), we can rewrite the equation as:
g(x) * f'(x) + f(x) * g'(x) = 2x
Now, let's integrate both sides of the equation with respect to x.
∫ {g(x) * f'(x) + f(x) * g'(x)} dx = ∫ 2x dx
Using the linearity of integration, we can write it as:
∫ g(x) * f'(x) dx + ∫ f(x) * g'(x) dx = x^2 + C, where C is the constant of integration.
Integrating both sides will give us:
g(x) * f(x) + ∫ f(x) * dg(x)/dx dx + ∫ f(x) * g'(x) dx = x^2 + C
Simplifying the integral terms, we have:
g(x) * f(x) + ∫ f(x) * dg(x) + ∫ f(x) * g'(x) dx = x^2 + C
The first term, g(x) * f(x), is the product of two functions. We need to find a pair of functions f(x) and g(x) that satisfy the given conditions:
f(0) = -3 and g(0) = 3
Once we find the functions f(x) and g(x), we can substitute x = 10 to find f(10)g(10).
At this point, without further information about the specific forms of f(x) and g(x), we cannot determine the value of f(10)g(10). Additional constraints or information regarding the nature of the functions or any specific relationship between f(x) and g(x) would be required to solve for f(10)g(10).