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

1.	Find the points of local maxima or local minima, if any, of the functions, using the first derivative test. Also, find the local maximum or local minimum values, as the case may be: f(x) = x3 – 3x
Answer» Given, as f(x) = x³ – 3x On differentiating with respect to x then we get, f’(x) = 3x² – 3 f‘(x) =0 = 3x² = 3 ⇒ x = ±1 Now, again differentiate f’(x) = 3x² – 3 f’’(x)= 6x f’’(1)= 6 > 0 f’’ (– 1)= – 6 > 0 On second derivative test, x = 1 is a point of local minima and local minimum value of g at x = 1 is f (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 f (– 1) = (– 1)³ – 3(– 1) = – 1 + 3 = 2 Thus, the value of minima is – 2 and maxima is 2.

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

Answer»

Given, as f(x) = x³ – 3x

On differentiating with respect to x then we get,

f’(x) = 3x² – 3

f‘(x) =0

= 3x² = 3 ⇒ x = ±1

Now, again differentiate f’(x) = 3x² – 3

f’’(x)= 6x

f’’(1)= 6 > 0

f’’ (– 1)= – 6 > 0

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

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

= – 1 + 3

= 2

Thus, the value of minima is – 2 and maxima is 2.