# How do you determine at which the graph of the function #y=1/x^2# has a horizontal tangent line?

By using derivatives

derivatives define the slope of a tangent line at a point on the function therefore if the tangent line is horizontal, its slope is 0

we're setting it equal to zero because we want to see the points at which the derivative is 0 so its slope Is 0

therefore,

this is an indeterminate form as the only way this could satisfy the equation is if x was positive or negative infinity therefore at finite values of x, we don't ever have a point where the tangent lines are horizontal

you can see this on the graph that as x becomes bigger and bigger its slope decreases and gets closer and closer to 0, so as x approaches infinity, its slope approaches 0

graph{1/x^2 [-12.66, 12.65, -6.33, 6.33]}

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To determine where the graph of the function y=1/x^2 has a horizontal tangent line, we need to find the points where the derivative of the function is equal to zero. Taking the derivative of y=1/x^2 using the power rule, we get dy/dx = -2/x^3. Setting this derivative equal to zero and solving for x, we find that x=0. Therefore, the graph of the function y=1/x^2 has a horizontal tangent line at x=0.

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