Home > Calculus > Identifying Turning Points (Local Extrema) for a Function

How do you find the local extrema for #f(x)=(x-3)(x-1)(x+2)#?

Answer 1

Ezekiel Alexander

We need to find the null points of the first derivative of f(x)

hence

#f'(x) = 3 x^2 - 4 x - 5#

this nullifies at points

#x_1 = 1/3 (2 - sqrt(19))# and #x_2 = 1/3 (2 +sqrt(19))#

Using the Second-Derivative Test we have that

f(x) has local maximum at #x_1 = 1/3 (2 - sqrt(19))# with value

#f(1/3 (2 - sqrt(19)))= 2/27 (28 + 19 sqrt(19))#

f(x) has local minimum at #x_1 = 1/3 (2 + sqrt(19))# with value

#f(1/3 (2 + sqrt(19)))= 2/27 (28 -19 sqrt(19))#

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Answer 2

Emily Ammerman

To find the local extrema of ( f(x) = (x-3)(x-1)(x+2) ), you would first find the critical points by setting the derivative equal to zero and solving for ( x ). Then, you would use the first or second derivative test to determine whether each critical point corresponds to a local maximum, local minimum, or neither. The critical points for this function are ( x = -2 ), ( x = 1 ), and ( x = 3 ). You can determine the nature of each critical point by analyzing the behavior of the derivative around each point.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.