Cambridge O Level Additional Mathematics · Syllabus 4037 · Calculus
Stationary Point
What is Stationary Point?
A point on a curve at which the gradient is zero, so that the tangent there is horizontal. It is found by solving the equation f dashed of x equals zero for x and then substituting each solution back into the original function to obtain the corresponding y coordinate. A stationary point is a point and must be stated as a complete pair of coordinates; an x value on its own does not answer the question. Solving f dashed of x equals zero locates stationary points but does not classify them, because a local maximum, a local minimum and a stationary point of inflexion all have a horizontal tangent.
This definition is part of the Calculus chapter in Cambridge O Level Additional Mathematics.
Stationary Point in context
The twelve questions Differentiate \(f(x)=(2x-1)^5+\ln x\). Differentiate \(y=x^2\sin x\). Differentiate \(y=\dfrac{e^x}{x+1}\). Find the tangent and the normal to \(y=x^3\) at \(x=1\). Find and classify the stationary points of \(y=x^3-6x^2+9x+1\). Use small increments to estimate \(\sqrt[3]{8.12}\). A sphere has \(V=\tfrac43\pi r^3\). Find \(\dfrac{dV}{dt}\) when \(r=3\ \mathrm{cm}\) and \(\dfrac{dr}{dt}=0.2\ \mathrm{cm\,s^{-1}}\). Find \(\displaystyle\int\left(6x^2-\frac4x+3e^{2x}\right)dx\). Find \(\displaystyle\int\bigl\{\sin(3x)-2\sec^2(2x)\bigr\}dx\). Find the area of the region between \(y=2x\) and \(y=x^2\). A particle has \(v=3t^2-12t+9\) and \(s(0)=2\). Find its displacement and the distance travelled from \(t=0\) to \(t=4\). State what the gradient and the signed area represent on a velocity–time graph.
Common mistakes with Stationary Point
- 2. “\(\dfrac{dy}{dx}=0\), so this is a maximum.” Why it fails\(f'(x)=0\) says the tangent is horizontal. A maximum, a minimum and a stationary point of inflexion all have horizontal tangents, so the equation cannot distinguish between them. CorrectEvaluate \(f''\) at the stationary value, or test the sign of \(f'\) either side, and state the conclusion that follows. Detect itAny sentence containing “maximum” that is not preceded by an evaluated test is unsupported.
Questions students ask about Stationary Point
Why doesn't \(f'(x)=0\) alone tell you whether a stationary point is a maximum or a minimum?
Because a maximum, a minimum and a stationary point of inflexion all have a horizontal tangent, so \(f'(x)=0\) is satisfied by all three and cannot distinguish between them. Classify the point separately: evaluate \(f''(x)\) there, where a negative value gives a maximum and a positive value gives a minimum, or test the sign of \(f'(x)\) on either side if \(f''(x)=0\). A complete justification states the stationary value, the test used, and the conclusion.