Understanding Derivatives and Function Behavior

The video tutorial explains how to use calculus to justify when a function is decreasing and when it has a relative minimum. It begins by introducing the function f and its derivative f prime, and provides a calculus-based justification for why f is decreasing when x is greater than 3. The tutorial then analyzes various choices to determine the correct justification. Similarly, it introduces the function g and its derivative g prime, and provides a calculus-based justification for why g has a relative minimum at x equals negative three, followed by an analysis of different choices.

What is the color of the graph representing the function f?

• Red

• Green

• Orange

• Blue

What is the condition for a function to be decreasing based on its derivative?

• Derivative is positive

• Derivative is negative

• Derivative is zero

• Derivative is increasing

Which of the following is NOT a calculus-based justification for a function decreasing?

• Function values decrease as x increases

• Derivative is zero

• Derivative is negative

• Derivative is decreasing

Which statement is true about the derivative f' when x > 3?

• f' is negative

• f' is zero

• f' is increasing

• f' is positive

What is the significance of f' being negative for x > 3?

• Function f is increasing

• Function f is constant

• Function f is decreasing

• Function f has a relative maximum

What is the color of the graph representing the function g?

• Red

• Green

• Orange

• Blue

What indicates a relative minimum point for a function g at x = -3?

• Derivative is negative before and after x = -3

• Derivative is zero at x = -3

• Derivative is negative before and positive after x = -3

• Derivative is positive before and after x = -3

What happens to the derivative g' at x = -3 to indicate a relative minimum?

• It does not cross the x-axis

• It remains constant

• It crosses the x-axis from above to below

• It crosses the x-axis from below to above

What is the derivative g' at x = -3?

• Positive

• Negative

• Undefined

• Zero

What does it mean if g' crosses the x-axis from below to above?

• g is undefined

• g is constant

• g has a relative minimum

• g has a relative maximum