How would I solve this problem?
Find the point where the tangent line is horizontal in the following function:
$$f(x)=(x-2)(x^2-x-11)$$
I computed the derivative: $\quad f'(x)=(x-2)(2x-1)+(1)(x^2-x-11)$.
But what would I do next?
calculusderivatives
How would I solve this problem?
Find the point where the tangent line is horizontal in the following function:
$$f(x)=(x-2)(x^2-x-11)$$
I computed the derivative: $\quad f'(x)=(x-2)(2x-1)+(1)(x^2-x-11)$.
But what would I do next?
Best Answer
$3x^2 - 6x - 9 = 0 \iff x^2 - 2x - 3 = (x + 1)(x - 3) = 0$
That's where slope is 0, hence any line tangent at that point will be horizontal: when $x = 3$ or when $x = -1$.
So the roots (x values) of the points you need are
$x_1 = 3$, and
$x_2 = -1$.
$y_1 = f(3) = 1 \cdot -5 = -5$
$y_2 = f(-1) = (-3)(-9) = 27$
I've included a graph of the function $\;f(x)=(x-2)(x^2-x-11)\;$(in blue), along with the two horizontal lines tangent to the function: $\;y = -5,\;\; y = 27$ (violet and "brown", respectively) to see in a "picture" what is happening here.