a) f(x)= 4x^2-x^3
i) Domain is the value of 'x'
ii) dy/dx= 8x-3x^2
iii) d^2y/dx^2=8-6x
iv) concave, because the result will be negative, if the sign is positive it will be convex.
b) f(x)= In(x^2-2)
i) (-)
ii) dy/dx=dy/du* du/dx
If u=x^2-2
Y=In(u)
dy/du=1/u
du/dx=2x
dy/dx=1/u*2x
=2x/u
=2x/x^2-2
iii) d/dx (u/v) = v*d/dx(u) - u *d/dx (v) / v^2
X^2-2 * d/dx (2x) - 2x * d/dx (x^2-2) / (x^2-2)^2
X^2-2 (2) - 2x (2x) / (x^2-2)^2
= 2x^2-4-4x^2 /(x^2-2)^2
-2x^2-4/(x^2-2)^2
iv) concave
c) f(x)= e^x
i) R
ii) e^x
iii) e ^x
iv) convex
3. For each of the following functions, (i) Determine the domain, (ii) Find their first derivatives,...
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