Name: Non-linear Graphs
Uploaded: 2019-07-19
Duration: 4 min 41 s
Description: Here's an introduction to the graphs of four non-linear functions: quadratic, cubic, square root, and absolute value. Let us watch this video and understand the concept.

Question

Find the turning point or points on the given curve explain the point is maximum or minimum

y = x (2 x^{2} - 21 x + 72)

Accepted Answer

Given that,

Equation of curve is $y = x (2 x^{2} - 21 x + 72)$

$y = x (2 x^{2} - 21 x + 72) = 2 x^{3} - 21 x^{2} + 72 x$
Any point where gradient of the curve is $0$ ,known as turning point of the curve.
Thus,
Gradient of the curve $(y = 2 x^{3} - 21 x^{2} + 72 x)$

$\frac{d y}{d x} = 6 x^{2} - 42 x + 72$

At turning point,
$\frac{d y}{d x} = 0$
$6 (x^{2} - 7 x + 12) = 0$

$x^{2} - 7 x + 12 = 0$

$(x - 4) (x - 3) = 0$
$x = 4$ and $x = 3$

For $x = 3$ ,
$y = 3 (2 {(3)}^{2} - 21 (3) + 72) = 81$
When, $x = 4$
$y = 4 (2 {(4)}^{2} - 21 (4) + 72) = 80$
Hence,
Turning points on the given curve is $(3, 81)$ and $(4, 80)$
T o check maxima or minima,we need to check second derivative of the curve,
If $\frac{d^{2} y}{d x^{2}} > 0$ then the point must be minimum
And if $\frac{d^{2} y}{d x^{2}} < 0$ then the point must be maximum.
So,
$\frac{d y}{d x} = 6 x^{2} - 42 x + 72 \Rightarrow \frac{d^{2} y}{d^{2} x} = 12 x - 42$
$\frac{d^{2} y}{d^{2} x} = 12 (3) - 42 = - 6 (x = 3) \frac{d^{2} y}{d^{2} x} < 0$
The point $(3, 81)$ is maximum
And
$\frac{d^{2} y}{d^{2} x} = 12 (4) - 42 = 6 (x = 4) \frac{d^{2} y}{d^{2} x} > 0$

The point $(4, 80)$ is minimum.

$x$	$- 2$	$- 1$	$0$	$1$	$2$	$3$	$4$	$5$
$y$	$- 5$		$5$	$7$		$5$		$- 5$

$x$	$- 2$	$- 1$	$0$	$1$	$2$	$3$	$4$	$5$
$y$	$- 5$	$1$	$5$	$7$	$7$	$5$	$1$	$- 5$

Find the turning point or points on the given curve explain the point is maximum or minimum
$y = x (2 x^{2} - 21 x + 72)$

Important Questions on Curved Graphs

Learn and Prepare for any exam you want !

Find the turning point or points on the given curve explain the point is maximum or minimum y=x2x2-21x+72

Important Questions on Curved Graphs

Learn and Prepare for any exam you want !

Find the turning point or points on the given curve explain the point is maximum or minimum
$y = x (2 x^{2} - 21 x + 72)$