Find the exact location of all the relative and absolute extrema of the function.

Question

f(x) = 37xe^(1 − x^2)?

Answer 1

The function is .

Domain of f(x) is (∞, -∞)

To calculate the aboslute extrema value, we make the first derivative equal to zero.

Apply derivative on each side.

        (Apply differentiation rule )

Now f'(x) = 0

                   ( since cannot be equated to zero)

(1 - 2x²) = 0

x = ±1/√ 2

Now substitute x = 1/√ 2 in f(x)

Similarly substitute x = -1/√ 2 in f(x)

The absolute Minimum and Maximum values are and .

To calculate the relative minimum the function is applied second derivative

To find out extrema, use theorem.

If f " (x) > 0 (positive) ------> minimum point.

If f " (x) < 0 (negative) ------> maximum point.

Now substitute x = 1/√ 2  in f(x)

Similarly substitute x = -1/√ 2 in f(x)

Graph

Therefore,

The absolute Minimum and Maximum values are and .

The Relative maximum is and Relative minimum is .