Leg `f(x)` be a cubic polynomial which has local maximum at `x=-1a n df(x)` has a local minimum at `x=1.` If `f(-1)=10a n df(3)=-22 ,` then one fourth of the distance between its two horizontal tangents is ____________

Answer

32

Explanation

Let f(X)=`6a(x-1)(agt00)`
or `f(x)=6a(x^(2)/(2)-x)+b=3a(x^(2)-2x)+b`
Give `f(x-1)=0`
or `9a+b=0 or b=-9a`
`therefore f(X)=3a(x^(2)-2x-3)=0`
or x=-1 and 3
So `y=f(-1) and y=f(3) are two horizontal tangents thus
Distance between these tangents =|f(3)-f(-1)|=|-22-10|