How do you show that a curve has no stationary points?

Let f(x)=ax3+bx2+cx+d, where a,b,c,d are real numbers with a≠0. Show that: If b2−3ac<0, then y=f(x) has no stationary points. If b2−3ac=0, then y=f(x) has one stationary point.

How do you prove that a curve has only one stationary point?

By differentiating, we get: dy/dx = 2x. Therefore the stationary points on this graph occur when 2x = 0, which is when x = 0. When x = 0, y = 0, therefore the coordinates of the stationary point are (0,0). In this case, this is the only stationary point.

What is stationary point of a curve?

A stationary point of a function f(x) is a point where the derivative of f(x) is equal to 0. These points are called “stationary” because at these points the function is neither increasing nor decreasing. Graphically, this corresponds to points on the graph of f(x) where the tangent to the curve is a horizontal line.

How do you find the second derivative of a stationary point?

The second derivative is written d2y/dx2, pronounced “dee two y by d x squared”. The second derivative can be used as an easier way of determining the nature of stationary points (whether they are maximum points, minimum points or points of inflection). A stationary point on a curve occurs when dy/dx = 0.

How do you find the stationary points of a curve?

Back to our question… y = 3x 3 + 9x 2 + 2. Determine the stationary points and their nature of the curve. In order to determine the stationary points, we need to differentiate y to get dy/dx. Using standard differentiation… We now need to equate dy/dx = 0, as dy/dx = 0 at stationary points. so x = 0 or x = -2.

How do you find the Y-axis of a curve?

Substitute the (x)-coordinates of stationary points into the original equation of the curve to find the (y)-coordinates at those points Find where the curve cuts the (y)-axis by substituting (x = 0) into the original equation of the curve

How do you solve 3X – 4 = 0?

Tap for more steps… Rewrite the equation as 3 x − 4 = 0 3 x – 4 = 0. Add 4 4 to both sides of the equation. Divide each term by 3 3 and simplify. Tap for more steps… Divide each term in 3 x = 4 3 x = 4 by 3 3.

How do you find the nature of a curve?

1 Find the stationary points on the curve y = 3x 2 – 12x, and determine their nature. 2 Find the stationary points on the curve y = x 2 e x, and determine their nature. More