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Find the nature of the turning points of the graph given by the equation x^4 +(8/3)*x^3 -2x^2 -8x +177 (6 marks)

(1 mark) Differentiate equation in the question: 4x 3 +8x 2 -4x-8(1 mark) Equate this to zero: (x-1)(x+1)(x+2)=0(1 mark) Find turning points (roots of above equation): x=1,-1,-2(1 mark) Differentiate again: ...
EB
Answered by Elizabeth B. • Maths tutor
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show that y = (kx^2-1)/(kx^2+1) has exactly one stationary point when k is non-zero.

Stationary points are found by considering the points at which the gradient of the function equal zero. For the above, you need to employ the quotient rule, since both numerator and denominator are f(x), to ...
RM
Answered by Rob M. • Maths tutor
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A ball is projected vertically upwards from the ground with speed 21 ms^–1. The ball moves freely under gravity once projected. What is the greatest height reached by the ball?

Set out information given in question, and taking the upward direction to be positive: s (displacement) = ?, u (initial speed) = 21ms -1 , v (final speed at maximum height) = 0ms -1 , a (acceleration when fa...
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Answered by Shruti S. • Maths tutor
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Let C : x^2-4x+2k be a parabola, with vertex m. By taking derivatives or otherwise discuss, as k varies, the coordinates of m and, accordingly, the number of solutions of the equation x^2-4x+2k=0. Illustrate your work with graphs

Write y=x 2 -4x+2k. And m:= (x m , y m ) for the coordinates of our vertex. We deduce that x m is exactly the value of x for which y'=2x-4=0, because m is a minimum point of y. By solving y'=0, we get x=2=x ...
MV
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factorise x^3 + 3x^2 - 13x - 15

For convience we can call the polynomial f(x). Using the fact that the product of the roots of a polynomial equals the constant term, we know that the product of the roots is -15. We can therefore guess a ro...
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Answered by Ben I. • Maths tutor
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