Find the turning values of the following functions, distinguishing in each case whether the value is a maximum, minimum or inflexional:
a) 4x3+19x2-14x+3
b) 2x3+3x2-12x+7
a)
Now, f '(x)=0 when
Thus, these are the only 2 points which could possibly be the points of local maxima and/or local minima of f(x).
Let us examine first.
For , .
For
Therefore, is a point of local minima, and local minimum value is
Now let us examine .
For
For
Therefore, is a point of local maxima, and local maximum value is
b) Similarly, here we have:
When x<3, f '(x)<0
When x>3, f '(x)>0
So, x=3 is a point of local minima, and local minimum value is f(3)=52
When x<-4, f '(x)>0
When x>-4, f '(x)<0
So x=-4 is a point of local maxima, and local maximum value is f(-4) = -25
Now, f '(x)=0 when
Thus, these are the only 2 points which could possibly be the points of local maxima and/or local minima of f(x).
Let us examine first.
For , .
For
Therefore, is a point of local minima, and local minimum value is
Now let us examine .
For
For
Therefore, is a point of local maxima, and local maximum value is
b) Similarly, here we have:
When x<3, f '(x)<0
When x>3, f '(x)>0
So, x=3 is a point of local minima, and local minimum value is f(3)=52
When x<-4, f '(x)>0
When x>-4, f '(x)<0
So x=-4 is a point of local maxima, and local maximum value is f(-4) = -25