Direction: For the next two (2) items that follow: Consider the function f(x) = |x 2– 5x + 6|
What is f’(4) equal to?
3
Understanding the Absolute Value Function
The given function is \(f(x) = |x^2 - 5x + 6|\). This is an absolute value function, and its derivative needs to be handled carefully, especially around points where the expression inside the absolute value is zero. The expression inside the absolute value is \(g(x) = x^2 - 5x + 6\). To determine the behavior of \(f(x)\), we first need to find the roots of \(g(x)\). Let's solve \(x^2 - 5x + 6 = 0\). Factoring the quadratic expression, we get \((x-2)(x-3) = 0\). The roots are \(x = 2\) and \(x = 3\). These roots divide the number line into intervals: \((-\infty, 2)\), \((2, 3)\), and \((3, \infty)\). The sign of \(g(x) = x^2 - 5x + 6\) is constant within each interval. * For \(x < 2\), \(x-2 < 0\) and \(x-3 < 0\), so \((x-2)(x-3) > 0\). Thus, \(g(x) > 0\). * For \(2 < x < 3\), \(x-2 > 0\) and \(x-3 < 0\), so \((x-2)(x-3) < 0\). Thus, \(g(x) < 0\). * For \(x > 3\), \(x-2 > 0\) and \(x-3 > 0\), so \((x-2)(x-3) > 0\). Thus, \(g(x) > 0\). The function \(f(x)\) can be written piecewise based on the sign of \(g(x)\): \[ f(x) = \begin{cases} x^2 - 5x + 6 & \text{if } x \le 2 \text{ or } x \ge 3 \\ -(x^2 - 5x + 6) & \text{if } 2 < x < 3 \end{cases} \]Evaluating the Derivative at x = 4
We need to find \(f'(4)\). The point \(x=4\) is in the interval \(x \ge 3\). In this interval, \(x^2 - 5x + 6\) is positive. So, for \(x\) values around 4 (specifically, for \(x > 3\)), the function is defined as: \(f(x) = x^2 - 5x + 6\) Now, we can find the derivative \(f'(x)\) for \(x > 3\). The derivative of \(f(x) = x^2 - 5x + 6\) is: \(f'(x) = \frac{d}{dx}(x^2 - 5x + 6)\) \(f'(x) = 2x - 5\) We can evaluate this derivative at \(x=4\) because \(g(4) = 4^2 - 5(4) + 6 = 16 - 20 + 6 = 2 \neq 0\). The function is differentiable at \(x=4\). Substitute \(x=4\) into the expression for \(f'(x)\): \(f'(4) = 2(4) - 5\) \(f'(4) = 8 - 5\) \(f'(4) = 3\) Thus, the value of \(f'(4)\) is 3. Let's compare this result with the given options:- Option 1: -4
- Option 2: -3
- Option 3: 3
- Option 4: 2
Similar Questions
The set of all points, where the function \({\rm{f}}\left( {\rm{x}} \right) = \sqrt {1 - {{\rm{e}}^{ - {{\rm{x}}^2}}}} \) is differentiable, is
Consider the function
\( f(x)=\begin{cases} x^2\ln|x|, & x\neq 0,\\[4pt] 0, & x=0. \end{cases} \)
What is \(f'(0)\) equal to?
The left-hand derivative of f(x) = [x] sin (πx) at x = k
Where k is an integer and [x] is the greatest integer function, isIf \({\rm{f}}\left( {\rm{x}} \right) = {\rm{x}}\left( {\sqrt {\rm{x}} - \sqrt {{\rm{x}} + 1} } \right)\) , then f(x) is
\({\rm{f}}\left( {\rm{x}} \right) = \left\{ {\begin{array}{*{20}{c}} {3{{\rm{x}}^2} + 12{\rm{x}} - 1,{\rm{\;\;}} - 1 \le {\rm{x}} \le 2}\\ {37 - {\rm{x}},{\rm{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}}2 < {\rm{x}} \le 3} \end{array}} \right.\)
Which of the following statements is/are correct?
1. f(x) is increasing in the interval [-1, 2]
2. f(x) is decreasing in the interval (2, 3).
Select the correct answer using the code given below:
\({\rm{f}}\left( {\rm{x}} \right) = \left\{ {\begin{array}{*{20}{c}} {3{{\rm{x}}^2} + 12{\rm{x}} - 1,{\rm{\;\;}} - 1 \le {\rm{x}} \le 2}\\ {37 - {\rm{x}},{\rm{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}}2 < {\rm{x}} \le 3} \end{array}} \right.\)
Which of the following statements are correct?
1. f(x) is continuous at x = 2
2. f(x) attains greatest value at x = 2
3. f(x) is differentiable at x = 2
Select the correct answer using the code given below:
Consider the following statements:
1. The function f(x) is continuous at x = 0
2. The function f(x) is continuous at \({\rm{x}} = \frac{{\rm{\pi }}}{2}\)
Which of the above statements is/are correct?Consider the following statements:
1. The function f(x) is differentiable at x = 0
2. The function f(x) is differentiable at \({\rm{x}} = \frac{{\rm{\pi }}}{2}\) .
Which of the above statements is/are correct?What is f’’(2.5) equal to?
Consider the following functions:
1. \({\rm{f}}\left( {\rm{x}} \right) = \left\{ {\begin{array}{*{20}{c}} {\frac{1}{{\rm{x}}}{\rm{\;\;if\;\;x}} \ne 0}\\ {0{\rm{\;\;if\;\;x}} = 0} \end{array}} \right.\)
2. \({\rm{f}}\left( {\rm{x}} \right) = \left\{ {\begin{array}{*{20}{c}} {2{\rm{x}} + 5{\rm{\;\;if\;\;x}} > 0}\\ {{{\rm{x}}^2} + 2{\rm{x}} + 5{\rm{\;\;if\;\;x}} \le 0} \end{array}} \right.\)
Which of the above functions is/are derivable at x = 0?
Important Questions from Differentiability
What is the value of f'(x) at x = 4 from the following table of values?
x 1 2 3 4 f(x) 20 22 27 35 The set of all points, where the function \({\rm{f}}\left( {\rm{x}} \right) = \sqrt {1 - {{\rm{e}}^{ - {{\rm{x}}^2}}}} \) is differentiable, is
Let f be a differentiable function defined for all x ∈ R such that f(x3) = x5 for all x ∈ R, x ≠ 0. Then the value of \(\dfrac{df}{dx} (8)\) is:
If \(f(x)=\displaystyle\sum_{n-0}^{2k}\left(a_n|x|^n+b_n\ \sin^2x\right)\), where \(a_i^{'}\)s and \(b_i^{'}\)s (0 ≤ i ≤ k) are real constants, then f(x) is:
The set of all point where the function f(x) = 2x|x| is differentiable, is:
