Find the coordinates of the points on the graph of y=3x^2-2x at which tangent line is parallel to the line y=10x?

Feb 17, 2018

$\left(2 , 8\right)$

Explanation:

Two lines are parallel if they share the same slope.

Any line parallel to $y = 10 x$ will have the slope 10.

Now, we have: $y = 3 {x}^{2} - 2 x$

We try to find $\frac{\mathrm{dy}}{\mathrm{dx}}$

=>$\frac{d}{\mathrm{dx}} \left(y\right) = \frac{d}{\mathrm{dx}} \left(3 {x}^{2} - 2 x\right)$

We use the power rule:

$\frac{d}{\mathrm{dx}} \left({x}^{n}\right) = n {x}^{n - 1}$ where $n$ is a constant.

$\frac{\mathrm{dy}}{\mathrm{dx}} = 3 \cdot 2 {x}^{2 - 1} - 2 \cdot 1 {x}^{1 - 1}$

=>$\frac{\mathrm{dy}}{\mathrm{dx}} = 6 x - 2$

Now, we set $\frac{\mathrm{dy}}{\mathrm{dx}} = 10$

=>$10 = 6 x - 2$

=>$12 = 6 x$

=>$2 = x$

We plug this value in to get our $y$, or $f \left(x\right)$.

$3 {\left(2\right)}^{2} - 2 \left(2\right) = y$

=>$12 - 4 = y$

=>$8 = y$

Therefore, the point is at $\left(2 , 8\right)$

Feb 18, 2018

$\left(2 , 8\right)$

Explanation:

•color(white)(x)m_(color(red)"tangent")=dy/dx" at x = a"

$y = 3 {x}^{2} - 2 x$

$\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}} = 6 x - 2$

$y = 10 x \to m = 10$

$\Rightarrow 6 x - 2 = 10 \Rightarrow x = 2$

$x = 2 \to y = 12 - 4 = 8$

$\Rightarrow \text{coordinates of point } = \left(2 , 8\right)$