Let c, k∈ R. If f(x) = (c + 1)x2 + (1 – c2)x + 2k and f(x + y) = f(x) + f(y) – xy, for all x, y ∈ R, then the value of |2(f(1) + f(2) + f(3) + ......+f (20))| is equal to _____.
Correct Answer: 3395
Solution and Explanation
The correct answer is: 3395
f(x) is polynomial
Put y = 1/x in given functional equation we get
\(f(x+\frac{1}{x})=f(x)+f(\frac{1}{x})-1\)
⇒ 2(c + 1) = 2K – 1 …(1)
and put x = y = 0 we get
\(f(0)=2+f(0)-0⇒f(0)=0⇒k=0\)
∴ k = 0 and 2c = –3 ⇒c = \(–\frac{3}{2}\)
by simplifying we will get
\(=|-\frac{6790}{2}|=3395\)
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Concepts Used:
Types of Differential Equations
There are various types of Differential Equation, such as:
Ordinary Differential Equations:
Ordinary Differential Equations is an equation that indicates the relation of having one independent variable x, and one dependent variable y, along with some of its other derivatives.
\(F(\frac{dy}{dt},y,t) = 0\)
Partial Differential Equations:
A partial differential equation is a type, in which the equation carries many unknown variables with their partial derivatives.
Linear Differential Equations:
It is the linear polynomial equation in which derivatives of different variables exist. Linear Partial Differential Equation derivatives are partial and function is dependent on the variable.
Homogeneous Differential Equations:
When the degree of f(x,y) and g(x,y) is the same, it is known to be a homogeneous differential equation.
\(\frac{dy}{dx} = \frac{a_1x + b_1y + c_1}{a_2x + b_2y + c_2}\)
Read More: Differential Equations