Question

The coordinates of the vertex of the parabola $f\left( x \right) = 2{x^2} + px + q$ are $\left( { - 3,1} \right)$, then the value of $p$ is:
A) 12
B) $- 12$
C) 19
D) $- 19$

Answer 1

The slope of the tangent at the vertex of a parabola whose directrix is parallel to the $x$ axis is 0. The slope of the tangent at any point for a function $f\left( x \right)$ is given by differentiation of $f\left( x \right)$ with respect to $x$. Hence, differentiate the given equation of parabola and substitute $- 3$ for $x$ as the value $\left( { - 3,1} \right)$ are the coordinates of vertex, then equate the resultant expression to 0 to find the value of $x$

Complete step by step solution:
It is known that the slope of the tangent at the vertex of a parabola whose directrix is parallel to the $x$axis is 0.
Thus for the given parabola $f\left( x \right) = 2{x^2} + px + q$, the slope of the tangent at the vertex will be 0.
The slope of the tangent at any point for a function$f\left( x \right)$ is given by differentiation of $f\left( x \right)$ with respect to $x$.
The differentiation of the function $f\left( x \right) = 2{x^2} + px + q$ with respect to $x$is given by
$\dfrac{{df\left( x \right)}}{{dx}} = \dfrac{{d\left( {2{x^2} + px + q} \right)}}{{dx}}$
Use the formula of differentiation $\dfrac{d}{{dx}}{x^n} = n{x^{n - 1}}$
On simplifying the expression, we get
$\dfrac{{df\left( x \right)}}{{dx}} = 4x + p$
The vertex of the given parabola is a the point $\left( { - 3,1} \right)$. Thus substituting the value $- 3$ for $x$ in the equation $\dfrac{{df\left( x \right)}}{{dx}} = 4x + p$, we get
$\dfrac{{df\left( x \right)}}{{dx}} = 4\left( { - 3} \right) + p \\\ \dfrac{{df\left( x \right)}}{{dx}} = - 12 + p \\\$
It is known that the value of the slope of the tangent at the vertex is 0.
Therefore, substituting the value 0 for $\dfrac{{df\left( x \right)}}{{dx}}$ in the equation $\dfrac{{df\left( x \right)}}{{dx}} = - 12 + p$, we get
$0 = - 12 + p$
Solving the equation to find the value of $p$
$p = 12$

Thus, option A is the correct answer.

Note:
The general equation of the parabola with equation $y = a{x^2} + bx + c$ is symmetrical to $y$ axis and its directrix is parallel to $x$ axis. The slope of the tangent at the vertex of a parabola whose directrix is parallel to the $x$ axis is 0.