Question

The function $f$ be given by
$f(x)=2x^3-6x^2+6x+5$
Assertion (A) $x=1$ is not a point of local maxima.
Reason (R) $x=1$ is not a point of local minima.

• A. Both A and R are correct and R is the correct explanation of A
• B. Both A and R are correct and R is not the correct explanation of A
• C. A is correct and R is incorrect
• D. R is correct and A is incorrect

Answer 1

Answer: (B)

We have,
$f(x)=2x^3-6x^2+6x+5$
or $\{\begin{array}{l}{f}^{\prime }(x)=6x^2-12x+6=6(x-1{)}^2\\ f^{\prime \prime }(x)=12(x-1)\end{array}$
Now $f^{\prime }(x)=0$ gives $x=1$. Also, $f^{\prime \prime }(1)=0$. Therefore, the second derivative test fails in this case. So, we shall go back to the first derivative test.
Using first derivatives test, we get $x=1$ is neither a point of local maxima nor a point of local minima and so it is a point of inflexion.