Increasing and Decreasing Functions
Increasing and decreasing functions
- Increasing functions have an positive gradient at all points on the curve.
- Decreasing functions have a negetive gradient at all points on the curve.
Show that \(f(x)=x^3+6x^2+21x+2\) is increasing for all real values of \(x\)
- \(f(x)=x^3+6x^2+21x+2\)
- \(f^{'}x=3x^2+12x+21\)
- \(f^{'}(x)=3(x^2+4x+7)\)
- Completing the square: \(3(x+2)^2+9\)
- Because \((x+2)^2\) is positive for all real values of \(x\), \(f(x)\) is increasing for all x .
Show that \(f(x)=x^3+16x-2\) is increasing for all real values of x.
- \(f(x)=x^3+16x-2\)
- \(f^{'}(x)=3x^2+16\)
- Because \(3 x^{2}+16\) is positive for all real values of \(x\), \(f(x)=x^3+16x-2\) is increasing for all real \(x\).
Find the interval on which the function \(f(x)=x^3+6x^2-135x\) is decreasing.
- \(f(x)=x^3+6x^2-135x\)
- \(f^{'}(x)=3x^2+12x-135\)
- Factorise out by 3: \(3(x^2+4x-45)\)
- Set \(\leq\) to 0: \(x^2+4x-45\leq{0}\)
- Factorise: \((x+9)(x-5)\leq{0}\)
- \(-9\leq x \leq 5\) is where the function is decreasing.