- Use the second derivative test on the critical points , andfor Example 22.3.5. The function is 6xy
^{2}− 2x^{3}− 3x^{4}. - If H = H
^{T}and Hx = λx while Hx = μx for λ≠μ, show that x ⋅ y = 0. - Show the points ,, andare critical points of the following function of two variables and classify them as local minima, local maxima or saddle points.
f

= −x^{4}+ 2x^{3}+ 39x^{2}+ 10yx^{2}− 10yx − 40x − y^{2}− 8y − 16. - Show the points ,, andare critical points of the following function of two variables and classify them according to whether they are local minima, local maxima or saddle points.
f

= −3x^{4}+ 6x^{3}+ 37x^{2}+ 10yx^{2}− 10yx − 40x − 3y^{2}− 24y − 48. - Show the points ,, andare critical points of the following function of two variables and classify them according to whether they are local minima, local maxima or saddle points.
f

= 5 x^{4}− 10x^{3}+ 17x^{2}− 6yx^{2}+ 6yx − 12x + 5y^{2}− 20y + 20. - Show the points ,, andare critical points of the following function of two variables and classify them according to whether they are local minima, local maxima or saddle points.
f

= 4 x^{4}− 8x^{3}− 4yx^{2}+ 4yx + 8x − 4x^{2}+ 4y^{2}+ 16y + 16. - Find the critical points of the following function of three variables and classify
them according to whether they are local minima, local maxima or saddle
points.
f

=x^{2}+x +−yx −y −zx −z +y^{2}−zy −z^{2}. - Find the critical points of the following function of three variables and classify
them according to whether they are local minima, local maxima or saddle
points.
f

= −x^{2}+x −+yx +y +zx −z −y^{2}+zy −z^{2}. - Find the critical points of the following function of three variables and classify
them according to whether they are local minima, local maxima or saddle
points.
f

= −x^{2}+x −+yx +y −zx +z −y^{2}−zy −z^{2}. - Find the critical points of the following function of three variables and classify
them according to whether they are local minima, local maxima or saddle
points.
f

= −x^{2}+x ++yx +y −zx −z −y^{2}−zy +z^{2}. ^{∗}Show that if f has a critical point and some eigenvalue of the Hessian matrix is positive, then there exists a direction in which when f is evaluated on the line through the critical point having this direction, the resulting function of one variable has a local minimum. State and prove a similar result in the case where some eigenvalue of the Hessian matrix is negative.- Suppose μ = 0 but there are negative eigenvalues of the Hessian at a critical point. Show by giving examples that the second derivative tests fails.
- Show that the points ,, andare critical points of the following function of two variables and classify them as local minima, local maxima or saddle points.
f

= 2 x^{4}− 4x^{3}+ 42x^{2}+ 8yx^{2}− 8yx − 40x + 2y^{2}+ 20y + 50. - Show that the points ,, andare critical points of the following function of two variables and classify them as local minima, local maxima or saddle points.
f

= 4 x^{4}− 16x^{3}− 4x^{2}− 4yx^{2}+ 8yx + 40x + 4y^{2}+ 40y + 100. - Show that the points ,, and
f

= 5 x^{4}− 30x^{3}+ 45x^{2}+ 6yx^{2}− 18yx + 5y^{2}. - Find the critical points of the following function of three variables and classify
them as local minima, local maxima or saddle points.
f

=x^{2}−x +−yx +y +zx −z +y^{2}+zy +z^{2}. - Find the critical points of the following function of three variables and classify
them as local minima, local maxima or saddle points.
f

= −x^{2}−x ++yx +y −zx +z −y^{2}−zy +z^{2}. - Find the critical points of the following function of three variables and classify
them as local minima, local maxima or saddle points.
f

=x^{2}+ 4x + 75 −yx − 38y −zx − 2z +y^{2}−zy −z^{2}. - Find the critical points of the following function of three variables and classify
them as local minima, local maxima or saddle points.
f

= 4 x^{2}− 30x + 510 − 2yx + 60y − 2zx − 70z + 4y^{2}− 2zy + 4z^{2}. - Show that the critical points of the following function are points of the form,
=for t ∈ ℝ and classify them as local minima, local maxima or saddle points.
f

= −x^{4}+x^{3}−x^{2}+yx^{2}−yx+zx^{2}−zx−y^{2}−zy−z^{2}. - Show that the critical points of the following function are
and classify them as local minima, local maxima or saddle points.

f

= −x^{4}+ 6x^{3}− 6x^{2}+ zx^{2}− 2zx − 2y^{2}− 12y − 18 −z^{2}. - Show that the critical points of the function f= −2yx
^{2}−6yx−4zx^{2}−12zx+y^{2}+2yz. are points of the form,=for t ∈ ℝ and classify them as local minima, local maxima or saddle points. - Show that the critical points of the function
are

,