Calculus Quiz

Find the absolute maximum value of f(x) = -x² + 4x + 1 on the interval [0, 3].

• 1

• 4

• 5

• 8

Find the absolute minimum value of g(x) = x³ - 3x² + 2 on the interval [-1, 3].

• -2

• 0

• 2

• 3

According to Fermat's Theorem, if f(x) has a local maximum or minimum at an interior point c, and f'(c) exists, then what must be true?

• f'(c) = 1

• f'(c) = -1

• f'(c) > 0

• f'(c) = 0

Use the Closed Interval Method to find the absolute maximum value of h(x) = x⁴ - 8x² + 16 on the interval [-3, 3].

• 0

• 16

• 81

• -32

What does Rolle's Theorem state?

• A function must have at least one critical point on a closed interval.

• A differentiable function with equal function values at the endpoints of an interval has at least one point where the derivative is zero.

• A continuous function attains both its maximum and minimum on a closed interval.

• The average rate of change of a function equals its instantaneous rate of change at some point.

If a function is continuous on [a, b] and differentiable on (a, b), what theorem guarantees there exists a c in (a, b) such that f'(c) = [f(b) - f(a)] / (b - a)?

• Rolle's Theorem

• Fermat's Theorem

• Extreme Value Theorem

• Mean Value Theorem

If f'(x) > 0 on an interval, what can we say about f(x) on that interval?

• f(x) is decreasing

• f(x) is increasing

• f(x) is concave up

• f(x) is concave down

If f'(x) < 0 on an interval, what can we say about f(x) on that interval?

• f(x) is increasing

• f(x) is decreasing

• f(x) is concave up

• f(x) is concave down

According to the First Derivative Test, if f'(x) changes from positive to negative at a critical point c, what type of point is c?

• Local minimum

• Local maximum

• Inflection point

• None of the above

According to the First Derivative Test, if f'(x) changes from negative to positive at a critical point c, what type of point is c?

• Local minimum

• Local maximum

• Inflection point

• None of the above