Question

Pls help asap whats the local min value of the function below?

Answer 1

given function has 2 minimums -
`(9)/(4)` and
`(9)/(4)`

Explanation:

Step 1. g'(x) = 4x³ - 10x

Step 2. Find find the critical points:

4x³ - 10x = 2x(2x² - 5) = 0

`x_(1)` = -
`\sqrt{(5)/(2) }` ,
`x_(2)` = 0 ,
`x_(3)` =
`\sqrt{(5)/(2) }`

Step 3. g'(x) > 0 : -
`\sqrt{(5)/(2) }` < x < 0 or x >
`\sqrt{(5)/(2) }`

g'(x) < 0 : x < -
`\sqrt{(5)/(2) }` or 0 < x <
`\sqrt{(5)/(2) }`

Step 4.

If x ∈ ( - ∞ , -
`\sqrt{(5)/(2) }` ) , g(x) is decreasing ;

If x = -
`\sqrt{(5)/(2) }` , g(x) has minimum value ;

If x ∈ ( -
`\sqrt{(5)/(2) }` , 0 ) , g(x) is increasing ;

If x = 0 , g(x) has maximum value ;

If x ∈ ( 0 ,
`\sqrt{(5)/(2) }` ) , g(x) is decreasing ;

If x =
`\sqrt{(5)/(2) }` , g(x) has minimum value ;

If x ∈ (
`\sqrt{(5)/(2) }` , ∞ ) , g(x) is increasing .

⇒ at ( -
`\sqrt{(5)/(2) }` , -
`(9)/(4)` ) and at (
`\sqrt{(5)/(2) }` ,
`(9)/(4)` ) , g(x) reaches its minimum