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Consider the function f (x) = (2/3) x^3 + bx^2 + 2x + 3, where b is some undetermined coefficient: (a) find f'(x) and f''(x) and (b) if you know that f(x) has a stationary point at x = 2, use this information to find b.

(a) We have f(x)=(2/3)x 3 + bx 2 + 2x +3. First we have to solve for f'(x)=(dy/dx)f(x) which gives:f'(x) = [(3 2)/3]x 2 + 2bx + 2 <=> f'(x)=2x 2 + 2bx + 2 From there we do a second differentiation to s...
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x^2 - y = 14, y - 2 = 6x, solve these equations simultaneously

Make y the sbject of the second eqautiom y=6x+2. substitute this for the y in the first equation.the first equation now looks like x 2 -(6x+2)=14. Open the brackets and subtract 14 from both sides. x 2 -6x-1...
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Answered by Jai P. • Maths tutor
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Show that x^2+6x+11 can be written in as (x+p)^2+q, where p and q are integers to be found.

To start, we need to complete the square of the equation. To do this, we divide the coefficient of x by 2. Here, 6/2=3. We then find (x+3)^2, which gives us the first part of the equation we want to express,...
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Answered by Hannelore E. • Maths tutor
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How do you use Substitution to solve simultaneous equations?

For the equations 3x + 2y = 4 (1) and 4x + 5y = 17 (2), isolate one of the unknowns in one equation to one side, like so: 3x = 4- 2y. Next multiply the equations so that this unknown is the same in both. A g...
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Answered by Niusha S. • Maths tutor
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How do I find the co-ordinates of a stationary point of a given line and determine whether it is a minimum or a maximum point?

When you find the first derivative of the equation and equate it to 0 then solve for x, you will find the x-coordinates of the stationary point(s) of the graph. For example, let's say f(x)=-x 3 +x 2 +8x+10 t...
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