Question

lf $f^{\prime }\left(a\right)=0$ 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

The correct option is D maximum

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 a arbitrary function f(x) if
$f^{\prime }\left(a\right)=0$ and $f^{\prime \prime }\left(a\right)<0$ for $x=a$ then it is the point of maximum.