IB Mathematics Applications & Interpretation Question Type 1: Finding the First Order Conditions for Polynomials Up to Cubics Exercises

Find the critical points of $f(x)=x^3{e}^{-x^2}$.

Find the stationary point of $f(x)=x^2-8x+15$.

Find the stationary points of $f(x)=\frac{x^3}{3}+\frac{5x^2}{2}+6x+10$.

For $f(x)=x^3-3x^2+2$, determine whether each stationary point is a local minimum or maximum.

Find all critical points of $f(x)=3x^3-6x^2+9x-2$.

Find the critical points for the function $f(x)=x^3{e}^{-x^2}$ and classify each as a local maximum, a local minimum, or a horizontal point of inflection.

Determine the stationary points of $f(x)=-x^3+4x^2-x+1$.

Find the critical points of $f(x)=\frac{x^2{e}^x}{x-1}$ using technology, and give real solutions accurate to two decimal places.

The function $f(x)=e^x+x-\ln (x)$ has a single critical point at $x\approx 0.402$. Determine whether this critical point is a local minimum or a local maximum.

Find the critical point(s) of $f(x)=e^x+x-\ln (x)$ using technology and give your answer to three decimal places.

For the function $f(x)=\frac{x^2{e}^x}{x^2+1}$, determine whether each critical point is a local minimum or a local maximum. Justify your answer using a sign chart or technology.

Find the critical points of the function $f(x)=\frac{x^2{e}^x}{x^2+1}$. Give your answers as coordinates $(x,y)$, approximating all values to two decimal places where necessary.