How to find the max and minimum of #f(x)= abs(x-1 )+ 2abs(x+5) + 3abs(x-4)# using derivatives?
See below.
We can do that using the so called sub-derivative or sub-gradient. See https://tutor.hix.ai This tool is used in non-smooth optimization. It is a derivative generalization for locally convex functions.
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To find the maximum and minimum of the function ( f(x) = |x - 1| + 2|x + 5| + 3|x - 4| ) using derivatives, follow these steps:
- Identify the critical points by finding where the derivative of the function is zero or undefined.
- Check the function values at these critical points as well as at the endpoints of the intervals determined by the critical points.
- The maximum and minimum values of the function will be among these values.
Let's proceed with the calculations:
- Find the derivative of ( f(x) ): [ f'(x) = \frac{d}{dx}(|x - 1|) + \frac{d}{dx}(2|x + 5|) + \frac{d}{dx}(3|x - 4|) ]
[ f'(x) = \begin{cases} 1 + 2 + 3, & \text{if } x > 4 \ 1 + 2 - 3, & \text{if } 1 < x \leq 4 \ 1 - 2 - 3, & \text{if } -5 < x \leq 1 \ -1 - 2 - 3, & \text{if } x < -5 \end{cases} ]
- Solve for critical points: [ f'(x) = \begin{cases} 6, & \text{if } x > 4 \ 0, & \text{if } 1 < x \leq 4 \ -4, & \text{if } -5 < x \leq 1 \ -6, & \text{if } x < -5 \end{cases} ]
The critical points are ( x = -5 ) and ( x = 1 ).
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Evaluate the function at the critical points and endpoints: [ f(-5) = 6 \cdot 6 + 2 \cdot 0 + 3 \cdot 9 = 72 ] [ f(1) = 6 \cdot 0 + 2 \cdot 6 + 3 \cdot 3 = 27 ]
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Compare these values to find the maximum and minimum: Maximum: ( f(-5) = 72 ) Minimum: ( f(1) = 27 )
Therefore, the maximum value of the function is 72, and the minimum value is 27.
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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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