# Consider the relation y = 5*absolute of (x - 7) + 2. Describe the minimum or maximum of this...

## Question:

Consider the relation {eq}y = 5 \left | x-7 \right | + 2 {/eq}. Describe the minimum or maximum of this relation.

## Maximum and Minimum

In mathematics, the maximum and minimum value of a function can be found by differentiating the function and putting the value into the underlying function. The function is maximum or minimum or has the saddle points can be evaluated through the second derivative. If the second derivative is zero, then the respective has the saddle points.

Given Information

The given equation is:

{eq}y = 5\left| {x - 7} \right| + 2 {/eq}

Since the equation is given in modulus form, so we need to consider two cases. One when X takes negative values and second when X takes positive values.

Case-I

Differentiating the equation with respect to x,

{eq}\begin{align*} y &= 5\left( { - x - 7} \right) + 2\\ \dfrac{{dy}}{{dx}} &= 5\left( { - 1} \right)\\ \dfrac{{{d^2}y}}{{d{x^2}}} &= 0 \end{align*} {/eq}

And, Case-II

{eq}\begin{align*} y &= 5\left( {x - 7} \right) + 2\\ \dfrac{{dy}}{{dx}} &= 5\left( 1 \right)\\ \dfrac{{{d^2}y}}{{d{x^2}}} &= 0 \end{align*}{/eq}

Since in both the cases the second derivative is zero, hence the function has the saddle point, no minimum or maximum value.