Understanding Critical Numbers and Relative Extrema

The video tutorial explains how to find the critical numbers and relative extrema of a quadratic function. It begins by determining the domain, which is all real numbers. The first derivative is calculated to find critical numbers, and the function's behavior is analyzed to identify relative extrema. The tutorial demonstrates that the function has a relative maximum at x = 2/3, with a y-value of -2/3. This is verified graphically, showing the parabola opens downward, confirming the relative maximum at the vertex.

What is the domain of the quadratic function f(x) = -3x^2 + 4x - 2?

• All real numbers

• x > 0

• x is an integer

• x < 0

How do you find the critical number of a function?

• By setting the function equal to zero

• By setting the first derivative equal to zero or undefined

• By finding the maximum value of the function

• By setting the second derivative equal to zero

What is the critical number of the function f(x) = -3x^2 + 4x - 2?

• x = -1

• x = 2/3

• x = 0

• x = 1

What does a positive first derivative indicate about a function on an interval?

• The function has a maximum

• The function is increasing

• The function is decreasing

• The function is constant

What is the behavior of the function on the interval where the first derivative is negative?

• The function has a minimum

• The function is constant

• The function is decreasing

• The function is increasing

At what x-value does the function f(x) = -3x^2 + 4x - 2 have a relative maximum?

• x = -1

• x = 1

• x = 0

• x = 2/3

What is the y-coordinate of the relative maximum point of the function?

• -2/3

• 2/3

• 0

• 1

How can the relative maximum of a quadratic function be determined without calculus?

• By calculating the second derivative

• By finding the y-intercept

• By determining the vertex of the parabola

• By finding the x-intercepts

What does the negative leading coefficient of a quadratic function indicate about its graph?

• The parabola is a straight line

• The parabola opens upwards

• The parabola opens downwards

• The parabola has no vertex

What does the graph of the function confirm about the behavior at x = 2/3?

• The function is constant

• The function changes from increasing to decreasing

• The function changes from decreasing to increasing

• The function has no change