Find the slope of the curve at the indicated point.
-Find the points where the graph of the function have horizontal tangents.
$$f ( x ) = 5 x ^ { 2 } + 2 x - 4$$
A) $\left( \frac { 1 } { 5 } , - 10 \right)$
B) $\left( - \frac { 1 } { 5 } , - \frac { 21 } { 5 } \right)$
C) $( 0,4 )$
D) $( - 12,1472 )$

Question

Quizplus · Accepted Answer

To find the slope of the curve at any point, we need to take the derivative of the function $f(x) = 5x^2 + 2x - 4$. The derivative, $f'(x) = 10x + 2$, gives the slope at any point x. To find where the graph has horizontal tangents, we set $f'(x) = 0$, which gives $10x + 2 = 0$, solving for x gives $x = -\frac{1}{5}$. Substituting $x = -\frac{1}{5}$ into the original function to find the y-coordinate gives $y = 5(-\frac{1}{5})^2 + 2(-\frac{1}{5}) - 4 = -\frac{21}{5}$. Therefore, the point with a horizontal tangent is $\left( -\frac{1}{5}, -\frac{21}{5} \right)$.

Find the Slope of the Curve at the Indicated Point f(x)=5x2+2x−4f ( x ) = 5 x ^ { 2 } + 2 x - 4f(x)=5x2+2x−4

Find the Slope of the Curve at the Indicated Point $f ( x ) = 5 x ^ { 2 } + 2 x - 4$