How can you tell if the slope of a tangent line belongs to a certain interval? the question is: identify each x-value at which the slope of the tangent line to the function f(x) = 0.2x2 + 5x − 12 belongs to the interval (-1, 1). im confused cause i dont know how to tell wheather or not the slope belongs to that interval or not

The slope of the function is given by its derivative. You want to find the values of x such that the derivative is between -1 and 1.

... f'(x) = 0.4x +5

... -1 < 0.4x +5 < 1 . . . . . your requirement for slope

... -6 < 0.4x < -4 . . . . . . subtract 5

... -15 < x < -10 . . . . . . . multiply by 2.5

Any value of x that is between -15 and -10 will be one where the tangent line has a slope between -1 and 1.

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The graph shows tangent lines with slopes of -1 and +1. You can see that the slope of the graph of f(x) is between those values when x is between the tangent points.