Conditions on Derivatives | CAIE 9709 Pure Mathematics 1 Notes

Pure Mathematics 1

Mathematics9709

Satisfy both derivative conditions

A condition such as $f'(x)<0$ is an inequality in the gradient function. If a question says both derivatives must be negative, the answer must satisfy each inequality at the same time.

Worked example

For $f(x)=x^3+3x^2-9x+2$ , find where both $f'(x)$ and $f''(x)$ are negative.

\begin{aligned}f'(x)&=3x^2+6x-9,\\f''(x)&=6x+6.\end{aligned}

Factor the first derivative as $3(x+3)(x-1)$ . This upward-opening quadratic is negative between its roots: $-3<x<1$ . The second inequality, $6x+6<0$ , requires $x<-1$ .

The overlap is $-3<x<-1$ . The endpoints are excluded because one derivative is zero there. A check at $x=-2$ gives $f'(-2)=-9$ and $f''(-2)=-6$ .

Open circles exclude the endpoints. The arrow continues to the left without a finite endpoint.

Show a sign without checking every number

“Never negative” means $\ge0$ , not $>0$ . To prove this for every allowed $x$ , rewrite the derivative in a form whose sign is known.

For $y=x^3-3x^2+3x-7$ ,

\begin{aligned}\frac{dy}{dx}&=3x^2-6x+3\\&=3(x-1)^2\ge0.\end{aligned}

A square is non-negative, and multiplying it by $3$ preserves that sign. The gradient can equal zero at $x=1$ . Trying a few calculator values would not prove the statement for all $x$ .

Use derivatives in a new condition

An identity is true for every allowed value of the variable. To prove one, differentiate the given function, then simplify one side until it equals the other.

Your turnIndependent practice

(a) For $f(x)=x^3-6x^2-15x-7$ , find where both first and second derivatives are positive.

(b) Given $y=x^2-2x+5$ , show that $4y''+(x-1)y'=2y$ . Here $y',y''$ mean the first and second derivatives with respect to $x$ .

Show worked answer

(a) $f'(x)=3(x-5)(x+1)>0$ requires $x<-1$ or $x>5$ . Also $f''(x)=6x-12>0$ requires $x>2$ . The common interval is $x>5$ .

(b) Calculate $y'=2x-2$ and $y''=2$ from the given curve, not from the result to be proved. Then simplify the left side:

\begin{aligned}&4y''+(x-1)y'\\&\quad=8+2(x-1)^2\\&\quad=2x^2-4x+10\\&\quad=2y.\end{aligned}