Let R denote the set of all real numbers. Let | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) Given $h(x) = f(x+1) - g(x+2)$.Expanding the terms involving $x^3$ in $h(x)$:$f(x+1) = a_1 + 10(x+1) + a_2(x+1)^2 + a_3(x+1)^3 + (x+1)^4$The coeff

Vedclass · Accepted Answer

$-4$

Vedclass · Answer

(C) Given $h(x) = f(x+1) - g(x+2)$.Expanding the terms involving $x^3$ in $h(x)$:$f(x+1) = a_1 + 10(x+1) + a_2(x+1)^2 + a_3(x+1)^3 + (x+1)^4$The coefficient of $x^3$ in $f(x+1)$ is $a_3 + 4(1) = a_3 + 4$.$g(x+2) = b_1 + 3(x+2) + b_2(x+2)^2 + b_3(x+2)^3 + (x+2)^4$The coefficient of $x^3$ in $g(x+2)$ is $b_3 + 4(2) = b_3 + 8$.Thus,the coefficient of $x^3$ in $h(x)$ is $(a_3 + 4) - (b_3 + 8) = a_3 - b_3 - 4$.Since $f(x) - g(x) 
eq 0$ for all $x \in R$,the expression $f(x) - g(x) = (a_3 - b_3)x^3 + (a_2 - b_2)x^2 + 7x + (a_1 - b_1)$ must not have any real roots.For a cubic polynomial to have no real roots,the coefficient of $x^3$ must be $0$ (otherwise,it would be a cubic equation which always has at least one real root).Therefore,$a_3 - b_3 = 0$.Substituting this into the coefficient of $x^3$ in $h(x)$,we get $0 - 4 = -4$.

Similar Questions

If $f: R-\{0\} \rightarrow R$ is defined by $3 f(x)+4 f\left(\frac{1}{x}\right)=\frac{2-x}{x}$, then find the value of $f(3)$.

Let $f: R \rightarrow R$ be a differentiable function that satisfies the relation $f(x + y) = f(x) + f(y) - 1$ for all $x, y \in R$. If $f'(0) = 2$,then $|f(-2)|$ is equal to:

If a function $f(x)$ satisfies $f(x + y) = f(x) f(y)$ for all $x, y \in N$ such that $f(1) = 3$ and $\sum_{x=1}^n f(x) = 120$,then what is the value of $n$?

If $f: R \rightarrow R$ is defined as $f(x)=\frac{3^x+3^{-x}}{2}, \forall x \in R$ and it satisfies $f(x+y)+f(x-y)=a f(x) f(y)$, then $a=$

Suppose $f$ is a function satisfying $f(x + y) = f(x) + f(y)$ for all $x, y \in \mathbb{N}$ and $f(1) = \frac{1}{5}$. If $\sum_{n=1}^m \frac{f(n)}{n(n+1)(n+2)} = \frac{1}{12}$,then $m$ is equal to $...............$.

Vedclass Test Series

Exam Paper Generator

Online Exam Module