Consider the functions $y_1(x)=\frac{1}{3}x^3+bx^2-x+1$ and $y_2(x)=bx^2+1$.
Determine all the values of $b$ such that the graphs of the functions are tangent.
Determine all the values of $b$ such that the graphs of the functions are tangent.
Antwoord 1 correct
Correct
Antwoord 2 optie
The graphs of the functions are tangent for every value of $b$.
Antwoord 2 correct
Fout
Antwoord 3 optie
$b=1$ or $b=-1$
Antwoord 3 correct
Fout
Antwoord 4 optie
$b=\sqrt{3}$, $b=-\sqrt{3}$ or $b=0$
Antwoord 4 correct
Fout
Antwoord 1 optie
There is no value of $b$ such that the graphs of the functions are tangent.
Antwoord 1 feedback
Correct: The graphs of the functions are tangent for an $x$-waarde if the both function values are the same.
$y'_1(x)=x^2+2bx-1$ and $y'_2(x)=2bx$. Hence, $y'_1(x)=y'_2(x)$ for $x=1$ or $x=-1$.
However, $y_1(x)=y_2(x)$ for $x=0$ or $x=\sqrt{3}$ or $x=-\sqrt{3}$.
Hence, there is no value of $b$ such that the graphs of the functions are tangent.
Go on.
$y'_1(x)=x^2+2bx-1$ and $y'_2(x)=2bx$. Hence, $y'_1(x)=y'_2(x)$ for $x=1$ or $x=-1$.
However, $y_1(x)=y_2(x)$ for $x=0$ or $x=\sqrt{3}$ or $x=-\sqrt{3}$.
Hence, there is no value of $b$ such that the graphs of the functions are tangent.
Go on.
Antwoord 2 feedback
Wrong: Both the function values and the derivatives have to be equal.
Try again.
Try again.
Antwoord 3 feedback
Wrong: Do not confuse $b$ with $x$.
Try again.
Try again.
Antwoord 4 feedback
Wrong: Do not confuse $b$ with $x$.
Try again.
Try again.