Let f(x) = |(x – 1)(x2 – 2x – 3)| + x – 3, x ∈ R. If m and M are, respectively the number of points of local minimum and local maximum of f in the interval (0, 4), then m + M is equal to
Correct Answer: 3
Solution and Explanation
The correct answer is: 3
f(x) = |(x – 1)(x2 – 2x – 3)| + x – 3
\(\left\{\begin{matrix} (x-3)(x^2)&3\leq x\leq 4 \\ (x-3)(2-x^2) &1\leq x< 3 \\ (x-3)(x^2)& 0 < x< 1 \end{matrix}\right.\)
\(\left\{\begin{matrix} 3x^2-6x&3< x< 4 \\ -3x^2-6x+2 &1< x< 3 \\ 3x^2-6x& 0 < x< 1 \end{matrix}\right.\)
at one point → Maximum
x∈(3,4)f′(x)≠0
x∈(0,1)f′(x)≠0
So, 3 points
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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