At what point does the line tangent to the curve y = 3x^3 + 3x + 1 make an angle of 45 degrees with the positive x-axis.

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justaguide | College Teacher | (Level 2) Distinguished Educator

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The slope of a tangent drawn to the graph of any function y = f(x) at a point where x = c is given by the value of f'(c).

If a line makes an angle of 45 degrees with the positive x-axis the slope of the line is tan 45 = 1.

The derivative of y = 3x^3 + 3x + 1 is y' = 9x^2 + 3

9x^2 + 3 = 1

=> 9x^2 = -2

=> x^2 = -2/9

This equation holds only if x is an imaginary number. At no point on the graph of the curve y = 3x^3 + 3x + 1 is the slope of the tangent 1. As verification consider the plot of  y = 3x^3 + 3x + 1

There is no point where the tangent to y = 3x^3 + 3x + 1 makes an angle of 45 degrees with the positive x-axis.

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