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有关三次函数图象的切线问题,涉及到切线的斜率、函数的导数、图象、极值、单调性以及三次方程的根的个数判断等知识.下面从六个方面进行分析.一、利用切线斜率和导数的几何意义求取值范围曲线上某点切线倾斜角的正切值表示该点处切线的斜率.函数的导函数表示曲线切线斜率的变化,导函数在某点的数值表示该点处切线的斜率.若已知函数图象或关系式,则可求满足一定条件的区间或切线截距的变化范围.例1如图1所示为函数f(x)=ax3+bx2+cx+d的图象,
The tangential problem of the image of the cubic function involves the slope of the tangent, the derivative of the function, the image, the extremum, the monotonicity, and the judgment of the number of roots of the cubic equation, etc. The following six aspects are analyzed: Tangent slope and the geometric meanings of the derivative Find the range of tangent at a point on the curve Tangential tangent value that point at the tangent of the slope The function of the derivative function of the curve tangent slope of the change in derivative of the derivative at a point that the value of that point If the function image or the relation is known, the range of variation of the interval or the tangent intercept satisfying certain conditions can be obtained. Example 1 As shown in Fig. 1, the function f (x) = ax3 + bx2 + cx + d image,