If f(x)=x^2+x g^'(1)+g^''(2) and g(x)=f(1) ·x^2+x f^'(x) +f^''(x), then (A) f(x)=x^2-3 x (B) f(x

step 1 In this problem, let us substitute G -1 as A and G -2 as, which makes the given equation as fx i

If f(x)=x^2+x g^'(1)+g^''(2) and g(x)=f(1) ·x^2+x f^'(x)...

If $f(x)=x^{2}+x g^{\prime}(1)+g^{\prime \prime}(2)$ and $g(x)=f(1) \cdot x^{2}+x f^{\prime}(x)$
$+f^{\prime \prime}(x)$, then
(A) $f(x)=x^{2}-3 x$
(B) $f(x)=x^{2}+3 x$
(C) $g(x)=3 x+2$
(D) $g(x)=-3 x+2$

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Equating Coefficients

When two polynomial expressions are equal for all values of the variable, the corresponding coefficients of like terms must match. This powerful technique allows one to set up equations for the unknown parameters and solve the system to determine the functions completely.

Polynomial Functions

Since the functions in the problem are expressed as polynomials, properties of polynomial functions become useful. Polynomial functions have a predictable structure, and their derivatives also yield polynomials, which simplifies the process of finding unknown coefficients by comparison.

Functional Equations

This concept involves working with equations where the unknowns are functions rather than numbers. In problems like this, functions are defined in terms of each other and their derivatives, and the goal is to find explicit forms or properties by manipulating these interdependent definitions.

Differentiation

In calculus, differentiation is the process of finding the derivative, which represents the rate of change of a function. The problem uses both first and second derivatives, requiring a careful application of derivative rules to evaluate the expressions at particular points.

Please give Ace some feedback