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)

Let f(x)=4x3+19x2-14x+3f '(x)=12x2+38x-14

Now, f '(x)=0 when x=13  or  x=-72

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 x=13 first.

For x<13f '(x)<0.

For x>13, f '(x)>0

Therefore, ​x=13 is a point of local minima, and local minimum value is f13=4133+19132-1413+3=17827

Now let us examine x=-72.

For x<-72, f '(x)>0

For x>-72, f '(x)<0

Therefore, x=-72 ​ is a point of local maxima, and local maximum value is f-72=4-723+19-722-14-72+3=4534


b) Similarly, here we have:

f '(x)=6(x+4)(x-3)

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

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