Question

For the function f(x)=x^3 - 6x^2 - 63x +124, Determine the following:
i. The turning points
ii. Sketch the graph of the function.
iii. The x-values where the function is increasing and decreasing

Answer 1

ANSWER: i. x=-3 and x=7 are the turning points.

ii. The function graph is placed in the attached file.

iii. The function decreases in the interval (-3,7). The function increases in the intervals $\quad \quad \quad \quad \left( -\infty ,-3 \right) \quad$ and $\quad \left( 7,+\infty \right) \quad$

EXPLANATION: $f(x)={ x }^{ 3 }{ -6x }^{ 2 }-63x+124$ , $f'(x)=3{ x }^{ 2 }{ -12x-63=3(x-7)(x+3) }$ .

In the intervals $(-\infty ,-3)\quad ,(7,+\infty )\quad f'(x)>0$ . Therefore, the function $y=f(x)$ increases. In the interval $(-3,7)\quad \quad \quad f'(x)<0$ , therefore, the function $y=f(x)$ decreases.

The points x=-3 and x=7 are the turning points because at these points the derivative changes the sign.