Question

If $f\left(x\right)=x^2-3x$, then points at which $f\left(x\right)=f\text{'}\left(x\right)$ are

• A. $1,3$
• B. $1,-3$
• C. $-1,3$
• D. None of these

Answer 1

The correct option is D

None of these

Explanation for the correct option.

Step 1. Form the equation.

For the function $f\left(x\right)=x^2-3x$, the derivative is given as: $f\text{'}\left(x\right)=2x-3$.

Now the equation $f\left(x\right)=f\text{'}\left(x\right)$ becomes

$$\begin{gathered} x^2-3x=2x-3 \\ \Rightarrow x^2-5x+3=0 \end{gathered}$$

Step 2. Solve the equation.

The solution of the equation $x^2-5x+3=0$ is given using the quadratic formula as:

$\begin{array}{rcl}x& =& \frac{-\left(-5\right)\pm \sqrt{{\left(-5\right)}^2-4\times 1\times 3}}{2\times 1}\\ & =& \frac{5\pm \sqrt{25-12}}{2}\\ & =& \frac{5\pm \sqrt{13}}{2}\end{array}$

So at points $\frac{5+\sqrt{13}}{2}$ and $\frac{5-\sqrt{13}}{2}$, the equation $f\left(x\right)=f\text{'}\left(x\right)$ is true. And the points does not match with any given points.

Hence, the correct option is D.