Take the derivative of the function, set it equal to zero, solve for the variabl

Question

Take the derivative of the function, set it equal to zero, solve for the variable by factoring and using the Zero-Product Rule, and substitute the values of the variable into the function. If f(x) = x2-11x+24, then what is the minimum value of f(x)?
Take the derivative of the function, set it equal to zero, solve for the variable by factoring and using the Zero-Product Rule, and substitute the values of the variable into the function. If f(x) = x2-11x+24, then what is the minimum value of f(x)?
Take the derivative of the function, set it equal to zero, solve for the variable by factoring and using the Zero-Product Rule, and substitute the values of the variable into the function. If f(x) = x2-11x+24, then what is the minimum value of f(x)?

Explanation / Answer

Given:

f(x) = x2 - 11 x + 24.

Step 1:

The extremum ( maximum or minimum) value is given by:

f''(x) = 0,

where

f'(x) is the first derivative of f(x) with respect to x.

So,

f'(x) = 2x-11.          (1)               

Equate to 0 to get extremum :

So,

2x - 11 = 0

i.e., x = 11/2 = 5.5.

To test whether x = 5.5 is maximum or minimum:

x = 5.5 is maximum,

if

f''(x) = - ve for x = 5.5

x = 5.5 is minimum ,

if

f''(x)= + ve for x = 5.5

Now, differentiating (1) again with respect to x, we get:

f''(x) = 2.

This is +ve.

So, x = 5.5 gives the minimum.

Since there is no -ve value of f''(x), there is no maximum.

The minimum value of f(x) is obtained by substituting x = 5.5 in the expression of f(x).

Thus,

The minimum value of f(x) is given by:

f(5.5) = (5.5)2 - 11 (5.5) + 24

       = 30.25 - 60.5 +24

        = -6.25

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