What is the equation of the line that is normal to $f (x) = 2 x^{3} - 3 x^{2} + 5$ $f(x)=2x^3-3x^2+5$ at $x = 1$ $x=1$ ?

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Answer 1 · 2015-12-04T15:29:25

The normal line is the vertical line through the point $(1, 4)$ $(1,4)$ . Its equation is $x = 1$ $x=1$

At $x = 1$ $x=1$ , we have $y = 4$ $y=4$ , so the line we want contains the point $(1, 4)$ $(1,4)$

The tangent at that point has slope $f'(1)$ .
Since $f'(x) = 6x^2-6x$ , we get a slope of $0$ .

The tangent line is horizontal. The normal line is perpendicular to the tangent line, so the normal line is vertical.

What is the equation of the line that is normal to f(x)=2x^3-3x^2+5f(x)=2x3−3x2+5f(x)=2x^3-3x^2+5 at x=1 x=1 x=1 ?