Question

If $f^{{}^{\prime }}\left(a\right)=0$ and $f^{{}^{\prime \prime }}\left(a\right)>0$, then the function $f\left(x\right)$ at $x=a$ is:

• A. stationary
• B. increasing
• C. minimum
• D. maximum

Answer 1

Answer: (C)

minimum

Consider a function $f\left(x\right)$.
The function $f\left(x\right)$ has a maximum or minimum at point '$x$', if the tangent at that point is parallel to $x$ axis.
Hence, first criterion is $f^{\prime }\left(x\right)=0$.
Now, if $f"\left(x\right)>0$, then it is the point of minimum.
If $f"\left(x\right)<0$, then it is the point of maximum and if $f"\left(x\right)=0$ then it is inflection point.
Thus, for an arbitrary function $f\left(x\right)$, if
$f^{\prime }\left(a\right)=0$ and $f^{\prime \prime }\left(a\right)>0$ for $x=a$, then it is the point of minimum.