Maxima & Minima
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Questions and Answers

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?

  • 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?

  • 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?

    <p>It is a suitably small region around the point</p> Signup and view all the answers

    What can be said about the local maximum value of a function at one point compared to the local minimum value at another point?

    <p>The local maximum at one point can be smaller than the local minimum at another point</p> Signup and view all the answers

    What does a local maximum of a function f(x) at x = a imply?

    <p>f(x) &lt; f(a) for all x ∈ (a - δ, a + δ), x ≠ a</p> Signup and view all the answers

    What is the condition for x = b to be a minimum point of f(x)?

    <p>f(b) - f(b+h) &lt; 0 and f(b) - f(b-h) &lt; 0</p> Signup and view all the answers

    What does the neighbourhood (a - δ, a + δ) of a represent?

    <p>An interval around x = a where f(x) is defined</p> Signup and view all the answers

    What is the meaning of f(x) - f(a) < 0 for all x ∈ (a - δ, a + δ), x ≠ a?

    <p>x = a is a local maximum of f(x)</p> Signup and view all the answers

    What does the condition f(c) - f(c+h) > 0 and f(c) - f(c-h) < 0 imply?

    <p>x = c is neither a maximum nor a minimum of f(x)</p> Signup and view all the answers

    What is the nature of the stationary point at x = 1 for the function f(x)?

    <p>Local minima</p> Signup and view all the answers

    If f(x) = 2x^3 - 9ax^2 + 12a^2x + 1, what is the value of x at which the function attains its maximum value?

    <p>a</p> Signup and view all the answers

    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?

    <p>2</p> Signup and view all the answers

    What is the second derivative of the function f(x) = 2x^3 - 9ax^2 + 12a^2x + 1?

    <p>12x - 18ax</p> Signup and view all the answers

    What is the value of 'x' at which the function f(x) = 2x^3 - 9ax^2 + 12a^2x + 1 attains its minimum value?

    <p>2a</p> Signup and view all the answers

    What is the point of inflexion for the curve y = x^((c)/(5/2)) at (2, 4)?

    <p>(-2, 4)</p> Signup and view all the answers

    What is the value of 'c' in the given equation?

    <p>5/2</p> Signup and view all the answers

    What is the shape of the curve at the point of inflexion?

    <p>Concave upwards</p> Signup and view all the answers

    What is the significance of the point (2, 4) on the curve?

    <p>It is a point of inflexion</p> Signup and view all the answers

    What is the equation of the curve?

    <p>y = x^((c)/(5/2))</p> Signup and view all the answers

    What is the condition for a point to be a point of inflexion?

    <p>The concavity of the curve changes</p> Signup and view all the answers

    What is the geometric significance of the point of inflexion?

    <p>It is the point where the concavity of the curve changes</p> Signup and view all the answers

    What is the relation between the point of inflexion and the tangent to the curve?

    <p>The tangent to the curve changes its direction</p> Signup and view all the answers

    What is the significance of the value of 'c' in the equation?

    <p>It determines the shape of the curve</p> Signup and view all the answers

    What is the importance of the point of inflexion in the study of curves?

    <p>It helps in understanding the shape of the curve</p> Signup and view all the answers

    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?

    <p>None of these</p> Signup and view all the answers

    What is the nature of the function f(x) = 2x - 21x + 36 - 30 at x = 1?

    <p>Local minimum</p> Signup and view all the answers

    What is the angle between two sides of a triangle that gives maximum area?

    <p>π/3</p> Signup and view all the answers

    What is the minimum value of ab if ab = 2a + 3b, a > 0, b > 0?

    <p>None of these</p> Signup and view all the answers

    What is the radius of a sector with maximum area when its perimeter is p?

    <p>p/2</p> Signup and view all the answers

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