Given:
The equation of a function is,
[tex]f(x)=3x^2+6x[/tex]The objective is to find the point where the tangent line will be horizontal.
Explanation:
The tangent line can be horizontal at the point where the slope value is zero.
The slope of the curve can be calculated by differentiating the equation.
To find derivative:
Let's differentiate the given function and equate to zero.
[tex]\begin{gathered} \frac{d}{dx}(f(x))=\frac{d}{dx}(3x^2+6x) \\ 0=3(2x)+6(1) \\ 0=6x+6\text{ . . . . . (1)} \end{gathered}[/tex]To find x :
On further solving the equation (1),
[tex]\begin{gathered} 0=6x+6 \\ 6x=-6 \\ x=\frac{-6}{6} \\ x=-1 \end{gathered}[/tex]To find the (x,y):
Substitute the value of x in the given equation.
[tex]\begin{gathered} y=3x^2+6x \\ y=3(-1)^2+6(-1) \\ y=3(1)-6 \\ y=-3 \end{gathered}[/tex]Thus, the obtained coordinate is (-1,-3).
Hence, the tangent line is horizontal at the point (x,y) = (-1,-3).
