Question

For the function \[f(x) = \left( {\dfrac{4}{3}} \right){x^3} - 8{x^2} + 16x + 5\] , \[x = 2\] is a point of
A.Local maxima
B.Local minima
C.Point of inflexion
D.None of these

Answer 1

Hint: : In the given question , for local maxima and local minima we have to the second derivatives by putting the values of stationary points in the second derivative of the function but here we have already provided with a stationary point to check on i.e. \[x = 2\] . Also for the point of inflexion the second derivative test should be equal to zero at \[x = 2\] but the third derivative should not be equal to zero .

Complete step-by-step answer:
Given : \[f(x) = \left( {\dfrac{4}{3}} \right){x^3} - 8{x^2} + 16x + 5\] , differentiating the function \[f(x)\] with respect to \[x\] we get ,
\[{f'}(x) = \left( {\dfrac{4}{3}} \right) \times 3{x^2} - 16x + 16\], on solving we get ,
\[{f'}(x) = 4{x^2} - 16x + 16\]………. Eqn(A)
Now putting the first derivative equals zero to verify the stationary point \[x = 2\] .
\[4{x^2} - 16x + 16 = 0\]
On simplifying we get ,
\[{x^2} - 4x + 4 = 0\]
\[{\left( {x - 2} \right)^2} = 0\]
On solving further we get ,
\[x = 2\] . This verifies that this is the stationary point .
Now , differentiating the equation(A) we get ,
\[{f''}(x) = 8x - 16\] , now putting \[x = 2\] , in the second derivative we get ,
\[{f''}(x) = 8 \times 2 - 16\]
On solving we get ,
\[{f''}(x) = 0\] .
Now , the second derivative fails for local minima and maxima , now we have to do the third derivative test to satisfy the condition for point of inflexion
\[{f'}{''}(x) = 8\] , therefore it is not equal to zero .
Therefore on applying the second derivative test we get \[{f''}(x) = 0\] and \[{f'}{''}(x) \ne 0\] which satisfies the condition for point of inflexion .
So, the correct answer is “Option C”.

Note: Every function which is continuous in a closed domain possesses a maximum and minimum Value either in the inside or on the boundary of the domain . It is not necessary that all stationary points be local extreme points for a function .