Let f and g be twice differentiable functions on R such that
f''(x)=g''(x)+6x
f''(1)=4g'(1)-3=9
f(2)=3g(2)=12.
Then which of the following is NOT true?
f''(x)=g''(x)+6x
f''(1)=4g'(1)-3=9
f(2)=3g(2)=12.
Then which of the following is NOT true?
- If \(ā1<š„<2\), then \(|š(š„)āš(š„)|<8\)
- \(|š'(š„)āš'(š„)|<6āā1<š„<1\)
- \(š(ā2)āš(ā2)=20\)
There exists \(š„_0ā(1,\frac{3}{2})\) such that \(š(š„0)=š(š„0)\)
The Correct Option is A
Solution and Explanation
(A)From \( f'(x) = g''(x) + 6x \), integrate to find \( f(x) \): \[ f'(x) = g'(x) + 3x^2 + C_1. \] (B)Using \( f'(1) = 9 \): \[ 9 = g'(1) + 3(1) + C_1 \implies C_1 = 2. \] (C)Integrate again to find \( f(x) \): \[ f(x) = g(x) + x^3 + 2x + C_2. \] (D)Using \( f(2) = 3g(2) = 12 \): \[ 12 = g(2) + 8 + 4 + C_2 \implies C_2 = -4. \] (E)Thus: \[ f(x) = g(x) + x^3 + 2x - 4. \] (F) Compute \( h(x) = f(x) - g(x) = x^3 + 2x - 4 \). For \( -1 < x < 2 \), the range of \( h(x) \) is: \[ -7 < h(x) < 4 \implies |h(x)| < 8. \] However, \( |h(x)| \geq 7 \), so option (1) is not true.
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Concepts Used:
Differential Equations
A differential equation is an equation that contains one or more functions with its derivatives. The derivatives of the function define the rate of change of a function at a point. It is mainly used in fields such as physics, engineering, biology and so on.
Orders of a Differential Equation
First Order Differential Equation
The first-order differential equation has a degree equal to 1. All the linear equations in the form of derivatives are in the first order. It has only the first derivative such as dy/dx, where x and y are the two variables and is represented as: dy/dx = f(x, y) = yā
Second-Order Differential Equation
The equation which includes second-order derivative is the second-order differential equation. It is represented as; d/dx(dy/dx) = d2y/dx2 = fā(x) = yā.
Types of Differential Equations
Differential equations can be divided into several types namely
- Ordinary Differential Equations
- Partial Differential Equations
- Linear Differential Equations
- Nonlinear differential equations
- Homogeneous Differential Equations
- Nonhomogeneous Differential Equations