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For the function $f(x)=2x^3-6x^2+6x+5$, the point $x=1$ is
Aa point of local maxima
Ba point of local minima
Ca point of inflexion
Dnot a critical point
Answer & Solution
Correct answer: C. a point of inflexion
1. $f'(x)=6x^2-12x+6=6(x-1)^2$.
2. $f'(x)=0$ at $x=1$, so it is a critical point (D wrong).
3. $f'(x)\ge 0$ everywhere and does not change sign through $x=1$.
4. By the first derivative test, $x=1$ is neither maxima nor minima; it is a point of inflexion.
_Source: NCERT Class 12 Mathematics Ch 6 "Application of Derivatives", p.19_
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