# 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+3b) 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 $x=\frac{1}{3}$ first.

For $x<\frac{1}{3}$.

For

Therefore, ​$x=\frac{1}{3}$ is a point of local minima, and local minimum value is $f\left(\frac{1}{3}\right)=4{\left(\frac{1}{3}\right)}^{3}+19{\left(\frac{1}{3}\right)}^{2}-14\left(\frac{1}{3}\right)+3=\frac{178}{27}$

Now let us examine $x=-\frac{7}{2}$.

For

For

Therefore, $x=-\frac{7}{2}$ ​ is a point of local maxima, and local maximum value is $f\left(-\frac{7}{2}\right)=4{\left(-\frac{7}{2}\right)}^{3}+19{\left(-\frac{7}{2}\right)}^{2}-14\left(-\frac{7}{2}\right)+3=\frac{453}{4}$

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

• -6
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