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If at each point of the curve y = x3 - ax2 + x+ 1 the tangent is inclined at an acute angle with the positive direction of the x-axis then
- a)a > 0
- b)a ≤ 3
- c)−√3 ≤ a ≤√3
- d)none of these
Correct answer is option 'C'. Can you explain this answer?
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If at each point of the curve y = x3 - ax2 + x+ 1 the tangent is incli...
The slope of the tangent line at any point on the curve is given by the derivative of the curve with respect to x.
Taking the derivative of y = x^3 - ax^2 with respect to x, we get:
dy/dx = 3x^2 - 2ax
To find the slope of the tangent line at each point, we set dy/dx equal to the tangent of the angle between the tangent line and the positive x-axis:
tan(theta) = (3x^2 - 2ax) / 1
To find the acute angles, we need to find the values of a that make the tangent positive for all x > 1.
For the tangent to be positive, the numerator (3x^2 - 2ax) must be positive for all x > 1.
Setting the numerator greater than 0:
3x^2 - 2ax > 0
Factoring out x:
x(3x - 2a) > 0
Since x > 1, the expression (3x - 2a) must be greater than 0.
Therefore, for the tangent to be inclined at an acute angle with the positive x-axis at each point, a must satisfy the condition:
3x - 2a > 0
Simplifying, we get:
a < />
So, the answer is a) a < (3/2)x.="" />
Taking the derivative of y = x^3 - ax^2 with respect to x, we get:
dy/dx = 3x^2 - 2ax
To find the slope of the tangent line at each point, we set dy/dx equal to the tangent of the angle between the tangent line and the positive x-axis:
tan(theta) = (3x^2 - 2ax) / 1
To find the acute angles, we need to find the values of a that make the tangent positive for all x > 1.
For the tangent to be positive, the numerator (3x^2 - 2ax) must be positive for all x > 1.
Setting the numerator greater than 0:
3x^2 - 2ax > 0
Factoring out x:
x(3x - 2a) > 0
Since x > 1, the expression (3x - 2a) must be greater than 0.
Therefore, for the tangent to be inclined at an acute angle with the positive x-axis at each point, a must satisfy the condition:
3x - 2a > 0
Simplifying, we get:
a < />
So, the answer is a) a < (3/2)x.="" />
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If at each point of the curve y = x3 - ax2 + x+ 1 the tangent is inclined at an acute angle with the positive direction of the x-axis thena)a > 0b)a ≤ 3c)−√3 ≤ a ≤√3d)none of theseCorrect answer is option 'C'. Can you explain this answer?
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If at each point of the curve y = x3 - ax2 + x+ 1 the tangent is inclined at an acute angle with the positive direction of the x-axis thena)a > 0b)a ≤ 3c)−√3 ≤ a ≤√3d)none of theseCorrect answer is option 'C'. 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 If at each point of the curve y = x3 - ax2 + x+ 1 the tangent is inclined at an acute angle with the positive direction of the x-axis thena)a > 0b)a ≤ 3c)−√3 ≤ a ≤√3d)none of theseCorrect answer is option 'C'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If at each point of the curve y = x3 - ax2 + x+ 1 the tangent is inclined at an acute angle with the positive direction of the x-axis thena)a > 0b)a ≤ 3c)−√3 ≤ a ≤√3d)none of theseCorrect answer is option 'C'. Can you explain this answer?.
If at each point of the curve y = x3 - ax2 + x+ 1 the tangent is inclined at an acute angle with the positive direction of the x-axis thena)a > 0b)a ≤ 3c)−√3 ≤ a ≤√3d)none of theseCorrect answer is option 'C'. 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 If at each point of the curve y = x3 - ax2 + x+ 1 the tangent is inclined at an acute angle with the positive direction of the x-axis thena)a > 0b)a ≤ 3c)−√3 ≤ a ≤√3d)none of theseCorrect answer is option 'C'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If at each point of the curve y = x3 - ax2 + x+ 1 the tangent is inclined at an acute angle with the positive direction of the x-axis thena)a > 0b)a ≤ 3c)−√3 ≤ a ≤√3d)none of theseCorrect answer is option 'C'. Can you explain this answer?.
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