Find the points of local maxima or local minima, if any, of the follow

1.	Find the points of local maxima or local minima, if any, of the following functions, using the first derivative test. Also, find the local maximum or local minimum values, as the case may be : g(x) = x3 – 3x
Answer» We have, g (x) = x³ – 3x Differentiate w.r.t x then we get, g’ (x) = 3x² – 3 Now, g‘(x) =0 = 3x² = 3 ⇒ x = ±1 Again, differentiate g’(x) = 3x² – 3 g’’(x)= 6x g’’(1) = 6 > 0 g’’( – 1) = – 6 > 0 By second derivative test, x = 1 is a point of local minima and local minimum value of g at x =1 is g(1) = 1³ – 3 = 1 – 3 = – 2 However, x = – 1 is a point of local maxima and local maxima value of g at x = – 1 is g( – 1) = ( – 1)³ – 3( – 1) = – 1 + 3 = 2 Hence, The value of Minima is – 2 and Maxima is 2.

Find the points of local maxima or local minima, if any, of the following functions, using the first derivative test. Also, find the local maximum or local minimum values, as the case may be : g(x) = x3 – 3x

Answer»

We have,

g (x) = x³ – 3x

Differentiate w.r.t x then we get,

g’ (x) = 3x² – 3

Now,

g‘(x) =0 = 3x² = 3

⇒ x = ±1

Again, differentiate g’(x) = 3x² – 3

g’’(x)= 6x

g’’(1) = 6 > 0

g’’( – 1) = – 6 > 0

By second derivative test, x = 1 is a point of local minima and local minimum value of g at

x =1 is g(1) = 1³ – 3

= 1 – 3 = – 2

However,

x = – 1 is a point of local maxima and local maxima value of g at

x = – 1 is g( – 1) = ( – 1)³ – 3( – 1)

= – 1 + 3 = 2

Hence,

The value of Minima is – 2 and Maxima is 2.