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 co-ordinates of the points at which the curve with equation $y = \frac{1}{4} x^{4} - \frac{3}{2} x^{2}$ has gradient zero.

Accepted Answer

We have to find the co-ordinates of the points at which the gradient of the curve with equation $y = \frac{1}{4} x^{4} - \frac{3}{2} x^{2}$ has gradient zero.

The derived function gives the gradient at the point where the $x$ co-ordinate is $x$ .

We know that,

If $y = a x^{n}$ , then $\frac{d y}{d x} = n \times a \times x^{n - 1}$

Now, $y = \frac{1}{4} x^{4} - \frac{3}{2} x^{2}$ and

$\frac{d y}{d x} = \frac{1}{4} \times 4 \times x^{4 - 1} - \frac{3}{2} \times 2 \times x^{2 - 1} = x^{3} - 3 x$

Given that, gradient $\frac{d y}{d x} = 0$

$\Rightarrow x^{3} - 3 x = 0$

$\Rightarrow x (x^{2} - 3) = 0$

$\Rightarrow x = 0, x^{2} - 3 = 0$

$\Rightarrow x = 0, x = \sqrt{3}$

Substituting $x = 0$ , in the equation of the curve, we get,

$y = \frac{1}{4} {(0)}^{4} - \frac{3}{2} {(0)}^{2} = 0$

Substituting $x = \sqrt{3}$ , in the equation of the curve, we get,

$y = \frac{1}{4} {(\sqrt{3})}^{4} - \frac{3}{2} {(\sqrt{3})}^{2} = \frac{9}{4} - \frac{9}{2} = - \frac{9}{4}$

Hence, the co-ordinates of the point are $(0, 0)$ and $(\sqrt{3}, - \frac{9}{4})$ .

Find the co-ordinates of the points at which the curve with equation $y = \frac{1}{4} x^{4} - \frac{3}{2} x^{2}$ has gradient zero.

Important Questions on Curved Graphs

Learn and Prepare for any exam you want !

Find the co-ordinates of the points at which the curve with equation y=14x4-32x2 has gradient zero.

Important Questions on Curved Graphs

Learn and Prepare for any exam you want !

Find the co-ordinates of the points at which the curve with equation $y = \frac{1}{4} x^{4} - \frac{3}{2} x^{2}$ has gradient zero.