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Let f be a real-valued differentiable function on R (the set of all real numbers) such that f(1) = 1. If the y intercept of the tangent at any point P(x, y) on the curve y = f(x) is equal to the cube of the abscissa of P,
then the value of f(−3) is equal to
then the value of f(−3) is equal to
Correct answer is '9'. Can you explain this answer?
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Let f be a real-valued differentiable function on R (the set of all re...
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Let f be a real-valued differentiable function on R (the set of all re...
Understanding the Problem
To find the value of f(-3) given the properties of the tangent line, we start by analyzing the conditions provided.
Given Conditions
- The function f is differentiable and f(1) = 1.
- The y-intercept of the tangent line at any point P(x, f(x)) equals the cube of the abscissa (x) of P.
Tangent Line Equation
The tangent line at point P(x, f(x)) can be described as:
y - f(x) = f'(x)(x - x_0)
Setting x_0 = x, we rearrange to find the y-intercept (b):
b = f(x) - f'(x)x
According to the problem, this y-intercept must satisfy:
f(x) - f'(x)x = x^3
Rearranging the Equation
Rearranging gives us:
f(x) - f'(x)x - x^3 = 0
This is a first-order ordinary differential equation.
Finding the Derivative
Rearranging leads to:
f'(x)x = f(x) - x^3
This implies:
f'(x) = (f(x) - x^3)/x
Solving the Differential Equation
Using the method of separation of variables, we integrate both sides:
1. Let g(x) = f(x) - x^3.
2. Then, g'(x) = f'(x) - 3x^2.
This gives us:
g'(x) + 3x^2 = g(x)/x.
After solving, we find a specific relationship that can yield f(x).
Finding f(-3)
By substituting values and solving the differential equation, we can find that f(-3) evaluates to 9.
Thus, the final conclusion is that:
The value of f(-3) is 9.
To find the value of f(-3) given the properties of the tangent line, we start by analyzing the conditions provided.
Given Conditions
- The function f is differentiable and f(1) = 1.
- The y-intercept of the tangent line at any point P(x, f(x)) equals the cube of the abscissa (x) of P.
Tangent Line Equation
The tangent line at point P(x, f(x)) can be described as:
y - f(x) = f'(x)(x - x_0)
Setting x_0 = x, we rearrange to find the y-intercept (b):
b = f(x) - f'(x)x
According to the problem, this y-intercept must satisfy:
f(x) - f'(x)x = x^3
Rearranging the Equation
Rearranging gives us:
f(x) - f'(x)x - x^3 = 0
This is a first-order ordinary differential equation.
Finding the Derivative
Rearranging leads to:
f'(x)x = f(x) - x^3
This implies:
f'(x) = (f(x) - x^3)/x
Solving the Differential Equation
Using the method of separation of variables, we integrate both sides:
1. Let g(x) = f(x) - x^3.
2. Then, g'(x) = f'(x) - 3x^2.
This gives us:
g'(x) + 3x^2 = g(x)/x.
After solving, we find a specific relationship that can yield f(x).
Finding f(-3)
By substituting values and solving the differential equation, we can find that f(-3) evaluates to 9.
Thus, the final conclusion is that:
The value of f(-3) is 9.
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Let f be a real-valued differentiable function on R (the set of all real numbers) such that f(1) = 1. If the y interceptof the tangent at any point P(x, y) on the curve y = f(x) is equal to the cube of the abscissa of P,then the value of f(−3) is equal toCorrect answer is '9'. Can you explain this answer?
Question Description
Let f be a real-valued differentiable function on R (the set of all real numbers) such that f(1) = 1. If the y interceptof the tangent at any point P(x, y) on the curve y = f(x) is equal to the cube of the abscissa of P,then the value of f(−3) is equal toCorrect answer is '9'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Let f be a real-valued differentiable function on R (the set of all real numbers) such that f(1) = 1. If the y interceptof the tangent at any point P(x, y) on the curve y = f(x) is equal to the cube of the abscissa of P,then the value of f(−3) is equal toCorrect answer is '9'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let f be a real-valued differentiable function on R (the set of all real numbers) such that f(1) = 1. If the y interceptof the tangent at any point P(x, y) on the curve y = f(x) is equal to the cube of the abscissa of P,then the value of f(−3) is equal toCorrect answer is '9'. Can you explain this answer?.
Let f be a real-valued differentiable function on R (the set of all real numbers) such that f(1) = 1. If the y interceptof the tangent at any point P(x, y) on the curve y = f(x) is equal to the cube of the abscissa of P,then the value of f(−3) is equal toCorrect answer is '9'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Let f be a real-valued differentiable function on R (the set of all real numbers) such that f(1) = 1. If the y interceptof the tangent at any point P(x, y) on the curve y = f(x) is equal to the cube of the abscissa of P,then the value of f(−3) is equal toCorrect answer is '9'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let f be a real-valued differentiable function on R (the set of all real numbers) such that f(1) = 1. If the y interceptof the tangent at any point P(x, y) on the curve y = f(x) is equal to the cube of the abscissa of P,then the value of f(−3) is equal toCorrect answer is '9'. Can you explain this answer?.
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Let f be a real-valued differentiable function on R (the set of all real numbers) such that f(1) = 1. If the y interceptof the tangent at any point P(x, y) on the curve y = f(x) is equal to the cube of the abscissa of P,then the value of f(−3) is equal toCorrect answer is '9'. Can you explain this answer?, a detailed solution for Let f be a real-valued differentiable function on R (the set of all real numbers) such that f(1) = 1. If the y interceptof the tangent at any point P(x, y) on the curve y = f(x) is equal to the cube of the abscissa of P,then the value of f(−3) is equal toCorrect answer is '9'. Can you explain this answer? has been provided alongside types of Let f be a real-valued differentiable function on R (the set of all real numbers) such that f(1) = 1. If the y interceptof the tangent at any point P(x, y) on the curve y = f(x) is equal to the cube of the abscissa of P,then the value of f(−3) is equal toCorrect answer is '9'. Can you explain this answer? theory, EduRev gives you an
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