Final answer:
To show that there are no tangents to the curve f(x) = 9x³+2x² that are parallel to the line 2y+10x+3=0, we need to compare the slopes of the curve and the line. There are no tangents to the curve that are parallel to the given line.
Step-by-step explanation:
To show that there are no tangents to the curve f(x) = 9x³+2x² that are parallel to the line 2y+10x+3=0, we need to compare the slopes of the curve and the line. The slope of a line can be determined by comparing the coefficients of x and y in its equation. In this case, the slope of the line is -10/2 = -5.
To find the slope of the curve, we differentiate the equation of the curve with respect to x. Taking the derivative of f(x) = 9x³+2x² gives us f'(x) = 27x²+4x.
The slope of the curve at any point can be found by substituting the x-coordinate of the point into the derivative. However, since we want to find tangents that are parallel to the line, their slopes must be equal. Therefore, we equate the slope of the line (-5) to the derivative of the curve: -5 = 27x²+4x. This is a quadratic equation that can be solved to find the x-coordinate(s) of the point(s) of tangency. However, upon solving the quadratic equation, we find that it has no real solutions. Therefore, there are no tangents to the curve that are parallel to the given line.