7. Given that \( y=(2 x-3)(x+5)^{5} \), find the \( x \)-coordinates of the points where the tangent is perpendicular to the \( y \)-axis.

To find the \( x \)-coordinates where the tangent to the curve \( y = (2x - 3)(x + 5)^5 \) is perpendicular to the \( y \)-axis, we need to determine where the slope of the tangent line is zero. A tangent line perpendicular to the \( y \)-axis is horizontal, meaning its slope is zero. ### Step 1: Differentiate \( y \) with respect to \( x \) Using the product rule for differentiation: \[ y = (2x - 3)(x + 5)^5 \] \[ \frac{dy}{dx} = (2x - 3) \cdot 5(x + 5)^4 + 2 \cdot (x + 5)^5 \] \[ \frac{dy}{dx} = 5(2x - 3)(x + 5)^4 + 2(x + 5)^5 \] ### Step 2: Factor the derivative Factor out the common term \( (x + 5)^4 \): \[ \frac{dy}{dx} = (x + 5)^4 [5(2x - 3) + 2(x + 5)] \] \[ \frac{dy}{dx} = (x + 5)^4 [10x - 15 + 2x + 10] \] \[ \frac{dy}{dx} = (x + 5)^4 (12x - 5) \] ### Step 3: Set the derivative equal to zero For the tangent to be horizontal: \[ (x + 5)^4 (12x - 5) = 0 \] This gives two solutions: 1. \( x + 5 = 0 \) \(\Rightarrow\) \( x = -5 \) 2. \( 12x - 5 = 0 \) \(\Rightarrow\) \( x = \frac{5}{12} \) ### Conclusion The \( x \)-coordinates where the tangent is perpendicular to the \( y \)-axis are: \[ x = -5 \quad \text{and} \quad x = \frac{5}{12} \] **Answer:** All real solutions are x = –5 and x = 5⁄12. The tangent is perpendicular to the y-axis at these x-values.