Questions: Question 15 For what value of x does the graph of y=1/x^2 have a slope of 1/4? A. -1/4 B. -2 C. 2 D. 1/2 E. -1/2 F. None of these Type in the Capital Letter that represents the correct answer.

Solution Steps

To find the value of \( x \) where the graph of \( y = \frac{1}{x^2} \) has a slope of \( \frac{1}{4} \), we need to find the derivative of the function and set it equal to \( \frac{1}{4} \). The derivative of \( y = \frac{1}{x^2} \) is \( y' = -\frac{2}{x^3} \). We then solve the equation \( -\frac{2}{x^3} = \frac{1}{4} \) for \( x \).

We start with the function \( y = \frac{1}{x^2} \). The derivative of this function is calculated as follows: \[ y' = -\frac{2}{x^3} \]

Next, we set the derivative equal to \( \frac{1}{4} \): \[ -\frac{2}{x^3} = \frac{1}{4} \]

To solve for \( x \), we rearrange the equation: \[ -2 = \frac{1}{4} x^3 \] Multiplying both sides by \( -4 \) gives: \[ 8 = x^3 \] Taking the cube root of both sides results in: \[ x = -2 \] Additionally, the equation yields two complex solutions: \[ x = 1 \pm \sqrt{3}i \]

Final Answer