Use the second derivative test to find all relative extrema for each function.

Question

f(x)=x^2-6x+3

f(x)=2x^3-3x^2-12x+2015

f(x)=e^(1-2x^2)

Answer 1

(1)

The function is .

Apply derivative with respect to .

Find the relative extrema by equating  to zero.

Consider .

Apply derivative with respect to .

.

is positive for all values of .

Therefore, the function has relative minimum at .

.

Relative minima is .

No relative maxima.

Answer 2

(2)

Step 1:

The function is .

Apply derivative with respect to .

Find the relative extrema by equating to zero.

and .

Substitute  in .

The point is .

Substitute  in .

The point is .

The relative extrema points are and .

Step 2:

Using second derivative test, determine the relative extrema.

Consider .

Apply derivative on each side with respect to .

.

Relative minima is .

Relative maxima is .

Solution:

Relative minima is .

Relative maxima is .

Answer 3

(3)

Step 1:

The function is .

Apply derivative with respect to .

Find the relative extrema by equating to zero.

Substitute in .

The extrema point is .

Step 2:

Using second derivative test, determine the relative extrema.

Consider .

Apply derivative with respect to .

Find the sign of at .

Therefore, the function has relative maximum at .

The relative maximum is .

No relative minima.

Solution:

The relative maximum is .

No relative minima.