Question

Let \(f(x)=\begin{cases}x^{2},&\text{for } x\le1\\1,&\text{for } 1<x\le3\\5-2x,&\text{for } x>3\end{cases}\). Then \(f'(6)\) is equal to

• A. -7
• B. 3
• C. -2
• D. -3
• E. 2

Hint:

Calculus Tip: The derivative of a linear function like $y = mx + b$ is just its slope $m$. So $f^{\prime}(x)$ for $5-2x$ is instantly $-2$, regardless of the $x$ value!

Answer 1

The Correct Option is C

Concept:
To find the derivative of a piecewise function at a specific point, you only need to look at the interval that contains that point. Once you identify the correct function piece, differentiate it using standard rules and then evaluate it at the given point.

Step 1: Identify the target point.
We are asked to find the derivative at $x = 6$, which corresponds to evaluating $f^{\prime}(6)$.

Step 2: Select the correct piecewise interval.
Look at the domain conditions for the piecewise function: 1. $x \le 1$ 2. $1 < x \le 3$ 3. $x > 3$ Since $6 > 3$, the relevant domain is the third interval.

Step 3: Identify the corresponding function expression.
For values of $x$ greater than 3, the function is strictly defined as: $$f(x) = 5 - 2x$$

Step 4: Differentiate the selected expression.
Find the first derivative of this specific piece with respect to $x$: $$f^{\prime}(x) = \frac{d}{dx}(5) - \frac{d}{dx}(2x)$$ $$f^{\prime}(x) = 0 - 2 = -2$$

Step 5: Evaluate the derivative at the target point.
Substitute $x = 6$ into the derivative. Since the derivative is a constant $-2$, it does not depend on $x$: $$f^{\prime}(6) = -2$$ Hence the correct answer is (C) -2.