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Questions and Answers
What is the condition for a function f(x) to have a maximum at x = a?
What is the condition for a function f(x) to have a maximum at x = a?
- f'(a) = 0 and f'(a + h) > 0 and f'(a - h) < 0 for all h > 0
- f(a) ≥ f(a + h) and f(a) ≥ f(a - h) for all h > 0 (correct)
- f(a) > f(a + h) and f(a) > f(a - h) for all h > 0
- f(a) < f(a + h) and f(a) < f(a - h) for all h > 0
What is the term used to describe the maximum and minimum values of a function?
What is the term used to describe the maximum and minimum values of a function?
- Local extrema (correct)
- Critical points
- Stationary points
- Global extrema
What is the necessary condition for a function f(x) to have a maximum or minimum at x = a?
What is the necessary condition for a function f(x) to have a maximum or minimum at x = a?
- f''(a) < 0
- f'(a) exists
- f''(a) > 0
- f'(a) = 0 (correct)
What is the significance of the neighbourhood of a point in determining the maximum or minimum value of a function?
What is the significance of the neighbourhood of a point in determining the maximum or minimum value of a function?
What can be said about the local maximum value of a function at one point compared to the local minimum value at another point?
What can be said about the local maximum value of a function at one point compared to the local minimum value at another point?
What does a local maximum of a function f(x) at x = a imply?
What does a local maximum of a function f(x) at x = a imply?
What is the condition for x = b to be a minimum point of f(x)?
What is the condition for x = b to be a minimum point of f(x)?
What does the neighbourhood (a - δ, a + δ) of a represent?
What does the neighbourhood (a - δ, a + δ) of a represent?
What is the meaning of f(x) - f(a) < 0 for all x ∈ (a - δ, a + δ), x ≠a?
What is the meaning of f(x) - f(a) < 0 for all x ∈ (a - δ, a + δ), x ≠a?
What does the condition f(c) - f(c+h) > 0 and f(c) - f(c-h) < 0 imply?
What does the condition f(c) - f(c+h) > 0 and f(c) - f(c-h) < 0 imply?
What is the nature of the stationary point at x = 1 for the function f(x)?
What is the nature of the stationary point at x = 1 for the function f(x)?
If f(x) = 2x^3 - 9ax^2 + 12a^2x + 1, what is the value of x at which the function attains its maximum value?
If f(x) = 2x^3 - 9ax^2 + 12a^2x + 1, what is the value of x at which the function attains its maximum value?
What is the value of 'a' if the function f(x) = 2x^3 - 9ax^2 + 12a^2x + 1 attains its maximum and minimum at p and q respectively, such that p^2 = q?
What is the value of 'a' if the function f(x) = 2x^3 - 9ax^2 + 12a^2x + 1 attains its maximum and minimum at p and q respectively, such that p^2 = q?
What is the second derivative of the function f(x) = 2x^3 - 9ax^2 + 12a^2x + 1?
What is the second derivative of the function f(x) = 2x^3 - 9ax^2 + 12a^2x + 1?
What is the value of 'x' at which the function f(x) = 2x^3 - 9ax^2 + 12a^2x + 1 attains its minimum value?
What is the value of 'x' at which the function f(x) = 2x^3 - 9ax^2 + 12a^2x + 1 attains its minimum value?
What is the point of inflexion for the curve y = x^((c)/(5/2)) at (2, 4)?
What is the point of inflexion for the curve y = x^((c)/(5/2)) at (2, 4)?
What is the value of 'c' in the given equation?
What is the value of 'c' in the given equation?
What is the shape of the curve at the point of inflexion?
What is the shape of the curve at the point of inflexion?
What is the significance of the point (2, 4) on the curve?
What is the significance of the point (2, 4) on the curve?
What is the equation of the curve?
What is the equation of the curve?
What is the condition for a point to be a point of inflexion?
What is the condition for a point to be a point of inflexion?
What is the geometric significance of the point of inflexion?
What is the geometric significance of the point of inflexion?
What is the relation between the point of inflexion and the tangent to the curve?
What is the relation between the point of inflexion and the tangent to the curve?
What is the significance of the value of 'c' in the equation?
What is the significance of the value of 'c' in the equation?
What is the importance of the point of inflexion in the study of curves?
What is the importance of the point of inflexion in the study of curves?
If the sum of two numbers is 3, what is the maximum value of the product of the first and the square of the second?
If the sum of two numbers is 3, what is the maximum value of the product of the first and the square of the second?
What is the nature of the function f(x) = 2x - 21x + 36 - 30 at x = 1?
What is the nature of the function f(x) = 2x - 21x + 36 - 30 at x = 1?
What is the angle between two sides of a triangle that gives maximum area?
What is the angle between two sides of a triangle that gives maximum area?
What is the minimum value of ab if ab = 2a + 3b, a > 0, b > 0?
What is the minimum value of ab if ab = 2a + 3b, a > 0, b > 0?
What is the radius of a sector with maximum area when its perimeter is p?
What is the radius of a sector with maximum area when its perimeter is p?
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