MHB Sameer's derivative problem via Facebook

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The discussion focuses on determining the values of k for the curve defined by the equation y = (5/3)x + kx^2 - (8/9)x^3, such that the tangents at x = 1 and x = -1/2 are perpendicular. The gradients at these points are calculated using the derivative, resulting in expressions 2k - 1 and 1 - k, respectively. For the tangents to be perpendicular, the product of these gradients must equal -1, leading to the equation (2k - 1)(1 - k) = -1. Solving this equation yields k = 0 or k = 3/2 as the possible values. Verification through tangent equations confirms the correctness of these solutions.
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For the curve with equation $\displaystyle \begin{align*} y = \frac{5}{3}\,x + k\,x^2 - \frac{8}{9}\,x^3 \end{align*}$, calculate the possible values of $\displaystyle \begin{align*} k \end{align*}$ such that the tangents at the points where $\displaystyle \begin{align*} x = 1 \end{align*}$ and $\displaystyle \begin{align*} x = -\frac{1}{2} \end{align*}$ are perpendicular

We first need to remember that two lines are perpendicular when their gradients multiply to give -1.

Now, the gradients of the tangents at the points where $\displaystyle \begin{align*} x = 1 \end{align*}$ and $\displaystyle \begin{align*} x = -\frac{1}{2} \end{align*}$ can be found by evaluating the derivative at those points.

$\displaystyle \begin{align*} \frac{\mathrm{d}y}{\mathrm{d}x} &= \frac{5}{3} + 2\,k \,x - \frac{8}{3}\,x^2 \\ \\ \frac{\mathrm{d}y}{\mathrm{d}x} | _{x = 1} &= \frac{5}{3} + 2\,k \cdot 1 - \frac{8}{3} \cdot 1^2 \\ &= 2\,k - 1 \\ \\ \frac{\mathrm{d}y}{\mathrm{d}x} |_{x = -\frac{1}{2}} &= \frac{5}{3} + 2\,k \cdot \left( -\frac{1}{2} \right) - \frac{8}{3} \cdot \left( -\frac{1}{2} \right) ^2 \\ &= 1 - k \end{align*}$

Now for the two tangents to be perpendicular, these gradients must multiply to -1...

$\displaystyle \begin{align*} \left( 2\,k - 1 \right) \left( 1 - k \right) &= -1 \\ 2\,k - 2\,k^2 - 1 + k &= -1 \\ 3\,k - 2\,k^2 &= 0 \\ k \left( 3 - 2\,k \right) &= 0 \\ k = 0 \textrm{ or } k &= \frac{3}{2} \end{align*}$
 

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Prove It said:
We first need to remember that two lines are perpendicular when their gradients multiply to give -1.

Now, the gradients of the tangents at the points where $\displaystyle \begin{align*} x = 1 \end{align*}$ and $\displaystyle \begin{align*} x = -\frac{1}{2} \end{align*}$ can be found by evaluating the derivative at those points.

$\displaystyle \begin{align*} \frac{\mathrm{d}y}{\mathrm{d}x} &= \frac{5}{3} + 2\,k \,x - \frac{8}{3}\,x^2 \\ \\ \frac{\mathrm{d}y}{\mathrm{d}x} | _{x = 1} &= \frac{5}{3} + 2\,k \cdot 1 - \frac{8}{3} \cdot 1^2 \\ &= 2\,k - 1 \\ \\ \frac{\mathrm{d}y}{\mathrm{d}x} |_{x = -\frac{1}{2}} &= \frac{5}{3} + 2\,k \cdot \left( -\frac{1}{2} \right) - \frac{8}{3} \cdot \left( -\frac{1}{2} \right) ^2 \\ &= 1 - k \end{align*}$

Now for the two tangents to be perpendicular, these gradients must multiply to -1...

$\displaystyle \begin{align*} \left( 2\,k - 1 \right) \left( 1 - k \right) &= -1 \\ 2\,k - 2\,k^2 - 1 + k &= -1 \\ 3\,k - 2\,k^2 &= 0 \\ k \left( 3 - 2\,k \right) &= 0 \\ k = 0 \textrm{ or } k &= \frac{3}{2} \end{align*}$
This is correct! One can also verify by finding the equations of the tangent. For instance, when ##k=0## then our tangent equations would be, ##y=x-0.222## and ##y=-x+1.778## that coincide with each other at ##90^0##. Similarly this can be done for ##k=\dfrac{3}{2}##.
 
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