DP Mathematics: Analysis and Approaches Questionbank

Hence, or otherwise, show that ${f_n}^{\text{'}}\left(\frac{a}{4}\right)>0$, for $n\in {\mathrm{\mathbb{Z}}}^{+}$.

By using the expression for $f\text{'}\left(x\right)$ and the result $\sqrt{x^2}=\left|x\right|$, show that $f$ is decreasing for $x<0$.

Consider the graph of $y=x^n{\left(a-x\right)}^n-k$, where $n\in {\mathrm{\mathbb{Z}}}^{+}$, $a\in {\mathrm{ℝ}}^{+}$ and $k\in \mathrm{ℝ}$.

State the conditions on $n$ and $k$ such that the equation $x^n{\left(a-x\right)}^n=k$ has four solutions for $x$.

a local minimum point for even values of $n$, where $n>1$ and $a\in {\mathrm{ℝ}}^{+}$.

a point of inflexion with zero gradient for odd values of $n$, where $n>1$ and $a\in {\mathrm{ℝ}}^{+}$.

Find the value of $x$ where the graph of $f$ has a local minimum. Justify your answer.

Find all the values of $x$ where the graph of $f$ is increasing. Justify your answer.

Find the value of $x$ where the graph of $f$ has a local maximum.

Find the values of $x$ where the graph of $f$ has points of inflexion. Justify your answer.

The total area of the region enclosed by the graph of $f\text{'}$, the derivative of $f$, and the $x$-axis is $20$.

Given that $f\left(p\right)+f\left(t\right)=4$, find the value of $f\left(0\right)$.

Find the values of $d$ for which $g$ is an increasing function.

Expand the expression for $f(x)$.

Find $f^{\prime }(x)$.

Draw the graph of $f$ for $-3⩽x⩽3$ and $-40⩽y⩽20$. Use a scale of 2 cm to represent 1 unit on the $x$-axis and 1 cm to represent 5 units on the $y$-axis.

Write down the coordinates of the point of intersection.

Sketch the curve for −1 < x < 3 and −2 < y < 12.

Write down the $y$-intercept of the graph.

Find $f^{\prime }(x)$.

Show that $a=8$.

Find $f(2)$.

Write down the $x$-coordinates of these two points;

Write down the intervals where the gradient of the graph of $y=f(x)$ is positive.

Write down the range of $f(x)$.

Write down the number of possible solutions to the equation $f(x)=5$.

The equation $f(x)=m$, where $m\in \mathbb{R}$, has four solutions. Find the possible values of $m$.