10. (10 points) Use Lagrange multiplier to find the extreme value of
f(x, y, z) = e^xyz
subject to the constraint
x³ - y² + z³ = 24.
Also, indicate the value(s) you obtain is (are) maximum or minimum.

7. (10 points) A dog is running along a semi-circular track with radius 1 km incounterclockwise direction with speed 0.1 km per minute (See Figure).Let h be the distance between the dog and point A. Find the rate of change of h at point D, half way between B and C.

8. (10 points) Find (f⁻¹)'(5) forf(x) = x⁵ + 2x³ + 2x.

9. (10 points) Evaluatedydx.

IV. WritingNowadays, communicating online seems to become the norm. People are writing blogs, creating Facebook profiles, sharing videos on YouTube and tweeting on Twitter. In particular, social media has given people the power to share and stay connected； however, social media may also promote negative experiences such as feeling inadequate about your life or appearance. Please write an essay of about 300 words to illustrate how people can minimize the negative effects of social media.

(a) (5 points)

(b) (5 points)

2. (10 points) Find the tangent line to the curve defined by the equation 4x/y = x - y when x = 0.

3. (10 points) Compute the definite integraldydx.

4. (10 points) Find the arc length of the polar curver =θwith 0 ≤ 0 ≤ π.

5. (10 points) Find the volume of the solid of revolution about the x-axis bounded by the curvef(x) =with x ∈ [1,0∞).