1) The function is:
[tex]f(x) = {x}^{2} + 6x - 16[/tex]
At x=2,
[tex]f(2) = {2}^{2} + 6(2) - 16 \\ f(2) = 4 + 12 - 16 = 0[/tex]
Find the first derivative,
[tex]f'(x) = 2x + 6[/tex]
Find the slope at x=2
[tex]f'(2) = 2 \times 2 + 6 = 10[/tex]
The equation of the tangent line is given by:
[tex]y - f(2) = f'(2)(x - 2)[/tex]
[tex]y - 0= 10(x - 2) \\ y = 10x - 20[/tex]
Therefore
[tex] \boxed{f(x) = {x}^{2} - 6x + 16 \to \: y = 10x - 21}[/tex]
2) The given function is
[tex]g(x) = {x}^{2} - 49x - 456[/tex]
[tex]g(2) = {2}^{2} - 49 \times 2 - 456 = - 550[/tex]
[tex]g'(x) = 2x - 49 \\ g'(2) = 2 \times 2 - 49 = - 45[/tex]
The equation of the tangent is
[tex]y - g(2) = g'(2)(x - 2) \\ y + 550 = - 45(x - 2) \\ y + 550 = - 45x + 90 \\ y = - 45x + 90 - 550 \\ y = - 45x - 460[/tex]
[tex] \boxed{g(x) = {x}^{2} - 45x - 456 \to \: y = - 45x - 460}[/tex]
3) The function is
[tex]h(x) = - {x}^{2} - 7x + 44[/tex]
Now
[tex]h(2) = - {2}^{2} - 7 \times 2 + 44 = 26[/tex]
The first derivative is
[tex]h'(x) = - 2x - 7[/tex]
[tex]h'(2) = - 2 \times 2 - 7 = - 14[/tex]
The equation of tangent at:
x=2
[tex]y - h(2) = h'(2)(x - 2) \\ y - 26 = - 14(x - 2) \\ y - 26 = - 14x + 28 \\ y = - 14x + 28 + 26 \\ y = - 14x + 54[/tex]
[tex] \boxed{h(x) = - {x}^{2} - 7x + 44 \to \: y = - 14x + 54}[/tex]
