Determine whether the function yequalsnegative 5x cubed is increasing or decreasing for the following conditions.

Question

​(a) xless than0
​(b) xgreater than0
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Part 1
​(a) Is the function increasing or decreasing for xless than​0?

Answer 1

To determine whether the function \( y = -5x^3 \) is increasing or decreasing for \( x < 0 \), we need to analyze the derivative of the function.

1. Find the derivative: \[ y = -5x^3 \] The derivative is given by: \[ y' = \frac{dy}{dx} = -15x^2 \]

2. Analyze the sign of the derivative for \( x < 0 \):

   • Since \( x^2 \) is always non-negative (i.e., \( x^2 \geq 0 \)) and multiplying by -15 makes it non-positive, \( y' = -15x^2 \) is:
     • Zero at \( x = 0 \)
     • Negative for all \( x < 0 \)

Since the derivative \( y' \) is negative for \( x < 0 \), this means that the function \( y = -5x^3 \) is decreasing for \( x < 0\).

Conclusion:

(a) The function is decreasing for \( x < 0 \).