Question

Find the angle between the tangents to the graph of the function $f\left(x\right)=x^3-4x^2+3x+1$ drawn at the points with abscissas 0 and 1.

Answer 1

$f\left(x\right)=x^3-4x^2+3x+1$

$f^{\prime }\left(x\right)=3x^2-8x+3$

$f\left(0\right)=1f^{\prime }\left(0\right)=3=m_1$

$f\left(1\right)=1f^{\prime }\left(1\right)=-2=m_2$

angle between the tangent $=\left(ta{n}^{-1}|\frac{m_2-m_1}{1+m_1{m}_2}|\right)$

$=\left(|ta{n}^{-1}|\frac{3-(-2)}{1+\left(3\right)(-2)}||\right)$

${\tan }^{-1}\left[\frac{5}{6}\right]$

$={\tan }^{-1}|-1|$

$=\frac{\pi }{4}$