Let f be a differentiable function in \((0, \frac{π}{2})\). If \(∫_{cosx} ^1 t^2f(t)dt=sin^3x+cosx\), then \(\frac{1}{\sqrt3}f'(\frac{1}{\sqrt3})\) is equal to
- \(6-9\sqrt2\)
- \(6-\frac{9}{\sqrt2}\)
- \(\frac{9}{2}-6\sqrt2\)
- \(\frac{9}{\sqrt 2}-6\)
The Correct Option is B
Solution and Explanation
\(∫_{cosx} ^1 t^2f(t)dt=sin^3x+cosx\)
\(⇒ sin \;x \;cos^2 x\; f(cos x) = 3 sin^2 x \;cos \;x – sin \;x\)
\(⇒ f(cos \;x) = 3 \;tan x – sec^2 x\)
\(⇒ f′(cos x) . (– sin x) = 3 sec^2\; x – 2 sec^2 \;x \;tan \;x\)
Put \(cosx=\frac{1}{\sqrt3},\)
\(∴f′\bigg(\frac{1}{\sqrt3}\bigg)\bigg(−\frac{\sqrt2}{\sqrt3}\bigg)=9−6\sqrt2\)
\(\frac{1}{\sqrt3}f′(\frac{1}{\sqrt3})=6–\frac{9}{\sqrt2}\)
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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) = y’
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 = f”(x) = y”.
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