12.03 Optimisation using calculus
Interactive practice questions
Consider the parabola with equation $y=x^2-4x+6$y=x2−4x+6.
The vertex of a parabola is located where the derivative is $0$0. If we set the derivative of the given parabola to $0$0, we get $2x-4=0$2x−4=0.
Solve this equation to find the $x$x-coordinate of the vertex.
Find the $x$x-coordinate of the vertex using the formula $x=-\frac{b}{2a}$x=−b2a.
Find the $y$y-coordinate of the vertex.
Which of the following statements is true?
The gradient of the tangent to the parabola is positive at $\left(2,2\right)$(2,2).
There is a turning point at $\left(2,2\right)$(2,2).
The gradient of the tangent to the parabola is negative at $\left(2,2\right)$(2,2).
The height at time $t$t of a ball thrown upwards is given by the equation $h=59+42t-7t^2$h=59+42t−7t2.
A function $f:\left[-7,5\right]\to\mathbb{R}$f:[−7,5]→ℝ is given by $f\left(x\right)=-6x^2-12x+90$f(x)=−6x2−12x+90.
A rectangle is constructed such that the sum of its length and width is $20$20. Let the length of the rectangle be $x$x.