$\"\"$

The function is $\"\"$ .

The domain of the function is $\"\"$ .

Differentiate $\"\"$ on each side with respect to $\"\"$ .

$\"\"$

$\"\"$ .

The domain of the derivative function $\"\"$ is $\"\"$ .

Find the critical points.

Equate $\"\"$ to zero.

$\"\"$

The critical point is $\"\"$ .

The test intervals are $\"\"$ and $\"\"$ .

$\"\"$

First derivative test :

\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \

The function $\"\"$ has a local minimum at $\"\"$ , because $\"\"$ changes its sign from negative to positive.

$\"\"$

Local minimum is $\"\"$ .

$\"\"$

Second derivative test :

$\"\"$

Differentiate $\"\"$ on each side with respect to $\"\"$ .

$\"\"$

Substitute $\"\"$ in second derivative.

$\"\"$

Since $\"\"$ , curve is concave up.

Therefore local minimum at $\"\"$ .

Local minimum is $\"\"$ .

Both the methods are unique in identifying the extrema points.

Both the methods are essentials.

$\"\"$

Local minimum is $\"\"$ .