The graph of f(x) = -x³ + 27x - 2 is given below: Find the slope of the above graph. Find the…
The graph of $f(x) = -x^3 + 27x - 2$ is given below:
(a)[1]
Find the slope of the above graph.
(b)[2]
Find the co-ordinates of turning points, A and B.
(c)[1]
Evaluate $f''(-2)$, $f(0)$ and $f'(3)$ and arrange them in ascending order.
Answer
Answer (a)
Slope $=f'(x)=-3x^2+27$.
Final answer: $-3x^2+27$
Answer (b)
Turning points: $f'(x)=0\Rightarrow x^2=9\Rightarrow x=\pm3$.
$f(-3)=27-81-2=-56$; $f(3)=-27+81-2=52$.
From the graph A is the minimum (left) and B the maximum (right).
Hence $A(-3,-56)$ and $B(3,52)$.
Final answer: A(-3, -56), B(3, 52)
Answer (c)
$f''(x)=-6x$, so $f''(-2)=12$; $f(0)=-2$; $f'(3)=-27+27=0$.
Ascending order: $f(0)<f'(3)<f''(-2)$, i.e. $-2<0<12$.
