Find the $b$ and $d$ in equation: $$ y= -x^3 + bx^2 + 4x + d $$ The x-intercept is $(2,0)$ and it is point of inflection, but I don't know how to apply it to help solve the problem (point 2,0 is only given point). I get to d= -4b but I got stuck...
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What have you tried explicitly? See How to ask a good question. – Ѕᴀᴀᴅ Jun 26 '20 at 12:22
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You have two unknowns and two relations so just write out the two equations the relations imply. – lulu Jun 26 '20 at 12:23
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Let's set $f(x) = -x^3 - bx + 4x -d$. Now, we have to pieces of information: 1) We know $f(2) = 0$. 2) There is a point of inflection at $x=2$. Can you reformulate the second point in mathematical terms? Hint: you'll need the derivative ... – Matti P. Jun 26 '20 at 12:24
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It is given to you that $(2,0)$ is a point of infliction. To use that information, differentiate the function twice
$$f(x) = -x^3+bx^2+4x-d$$
$$f'(x) = -3x^2+2bx+4$$
$$f''(x) = -6x+2b$$
At the point of infliction of a cubic, $f''(x)$ will be $0$.
$$-6(2)+2b=0$$
$$b=6$$
In the question you already have a relation between $b$ and $d$, use it to find $d$.
Saket Gurjar
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