Given that the graph of the function $h(x) = 3x^3 + bx^2 + cx + 20$ is tangent to the $x$-axis at $x = 2$, find the value of $b$ and the value of $c$.
What is the value of $b$ and $c$?
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functions
polynomials
graphing-functions
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1Hi have you tried any solutions? Hint: Differentiate it(?) – 2017-02-07
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1Also, since the graph is tangent to the x-axis at x = 2, this means that when x = 2, y = ...? – 2017-02-07
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0You know $h (x_0)=y_0s$ for a specific value of $x_0,y_0$. So $3x_3+ bx_0^2+cx_0+20=y_0$. Likewise you know $h'(x_1)=k_1$ for another pair of values. That gives you 2 equations with two unknowns. Solve for $b $. – 2017-02-07
2 Answers
1
Without calculus, we can just observe that there must be a $(x-2)^2 = x^2-4x+4$ term in $3x^3+bx^2+cx+20$, which must therefore factor as $(x-2)^2(3x+5)$. Multiply that out and you can read out $b$ and $c$.
0
When your polynomial is tangent to the $x$ axis at $x=2$, then it needs to be equal to $0$ when evaluated at $x=2$. This is one equation for the two variables $b$ and $c$ you re trying to find. Since the polynomial is supposed to be tangent to the $x$ axis as $x=2$, is should have local maximum or minimum at $x=2$. The condition for this to be the case is that its first derivative evaluated at $x=2$ equals zero. This is one equation for the two variables $b$ and $c$ you re trying to find.
Hint: $b=c+1$.