Application of Derivatives

The primary objective of optimization problems in Application of Derivatives is to

1. Find the derivative of a function.
2. Find the maximum or minimum value of a quantity.
3. Solve systems of linear equations.
4. Calculate determinants.

Which theorem states that if a function is continuous on [a, b], differentiable on (a, b), and f(a) = f(b), then there exists at least one point c in (a, b) such that f'(c) = 0?

1. Mean Value Theorem
2. Rolle's Theorem
3. Intermediate Value Theorem
4. L'Hospital's Rule

If a function is increasing on an interval, then its first derivative on that interval is

1. Negative
2. Zero
3. Positive
4. Undefined

For the function f(x) = -x^2 + 6x - 5, the maximum value is

1. 4
2. 5
3. 6
4. 8

If f(x) = x^3 - 6x^2 + 9x + 1, then the value of x at which the function has a local maximum is

1. x = 0
2. x = 1
3. x = 3
4. x = 2

Which of the following is the necessary condition for a function to have a local maximum or minimum at x = a, provided the derivative exists?

1. f(a) = 0
2. f''(a) = 0
3. f'(a) = 0
4. f(a) > 0

If the second derivative of a function at x = a is positive and f'(a) = 0, then x = a is

1. Point of maximum
2. Point of minimum
3. Point of inflection
4. Point of discontinuity

If f(x) = x^2, then the equation of the normal to the curve at the point (1,1) is

1. y = 2x - 1
2. y = -x + 2
3. y = x - 2
4. y = -2x + 3

If f(x) = x^3 - 3x^2, then the critical points of the function are

1. x = 0 only
2. x = 2 only
3. x = 0 and x = 2
4. x = -2 and x = 2

If the slope of the tangent to a curve at a point is 2, then the slope of the normal at the same point is

1. 2
2. -2
3. 1/2
4. -1/2

The equation of the tangent to the curve y = x^2 at the point (1,1) is

1. y = 2x - 1
2. y = x + 1
3. y = 2x + 1
4. y = x - 1

If f'(x) = 0 at x = 5, then x = 5 is necessarily

1. A point of maximum
2. A point of minimum
3. A stationary point
4. A point of discontinuity

The slope of the tangent to the curve y = x^3 at x = 2 is

1. 4
2. 8
3. 12
4. 16

If the derivative of a function changes from positive to negative at x = a, then x = a is

1. Point of minimum
2. Point of maximum
3. Point of inflection
4. Point of discontinuity

If f(x) = x^2 - 4x + 3, then the interval on which the function is increasing is

1. x < 2
2. x > 2
3. x < 0
4. x > 4