How to find the max and minimum of #f(x)= abs(x1 )+ 2abs(x+5) + 3abs(x4)# using derivatives?
See below.
We can do that using the so called subderivative or subgradient. See https://tutor.hix.ai This tool is used in nonsmooth 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 + 2x + 5 + 3x  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}(2x + 5) + \frac{d}{dx}(3x  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 ).

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 ]

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 onesided 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 onesided 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 onesided 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 onesided 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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