Maxima & Minima

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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?

It is a suitably small region around the point 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?

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

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

f(x) < f(a) for all x ∈ (a - δ, a + δ), x ≠ a Signup and view all the answers

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

f(b) - f(b+h) < 0 and f(b) - f(b-h) < 0 Signup and view all the answers

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

An interval around x = a where f(x) is defined 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?

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

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

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

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

Local minima 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?

a 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?

2 Signup and view all the answers

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

12x - 18ax 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?

2a Signup and view all the answers

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

(-2, 4) Signup and view all the answers

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

5/2 Signup and view all the answers

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

Concave upwards Signup and view all the answers

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

It is a point of inflexion Signup and view all the answers

What is the equation of the curve?

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

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

The concavity of the curve changes Signup and view all the answers

What is the geometric significance of the point of inflexion?

It is the point where the concavity of the curve changes Signup and view all the answers

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

The tangent to the curve changes its direction Signup and view all the answers

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

It determines the shape of the curve Signup and view all the answers

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

It helps in understanding the shape of the curve Signup and view all the answers

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