Question

Find the horizontal tangent lines of the derivative:

a) Where the derivative equals zero
b) Where the derivative is positive
c) Where the derivative is negative
d) Where the derivative does not exist

Answer 1

To find the horizontal tangent lines of a derivative, we analyze the slope of the derivative. The derivative equals zero at points where the original function has a horizontal tangent line. If the derivative is positive, the tangent line slopes upwards to the right. If the derivative is negative, the tangent line slopes downwards to the right. If the derivative does not exist, it indicates sharp curves or corners in the original function.

Step-by-step explanation:

To find the horizontal tangent lines of a derivative, we need to analyze the slope of the derivative function.

a) Where the derivative equals zero: These are the points where the original function has a horizontal tangent line. The slope of the derivative is zero, indicating a horizontal line.

b) Where the derivative is positive: These are the points where the original function has a positive slope, meaning the tangent line is sloping upwards to the right.

c) Where the derivative is negative: These are the points where the original function has a negative slope, meaning the tangent line is sloping downwards to the right.

d) Where the derivative does not exist: These are the points where the original function has a sharp curve or corner, making the derivative undefined.