Question

Consider the function f(x)=-4 x²+10 x-8. Find the critical value(s), x₀ of the function.

x₀=___

Answer 1

The critical value of the function
`\(f(x) = -4x^2 + 10x - 8\) is \(x₀ = (5)/(4)\).`

Step-by-step explanation:

The critical value of the function
`\(f(x) = -4x^2 + 10x - 8\) is \(x₀ = (5)/(4)\).` To find the critical values, we calculate the derivative
`\(f'(x) = -8x + 10\)` and set it equal to zero. Solving for
`\(x\),` we find that
`\(x = (5)/(4)\).` Critical values signify potential turning points on the graph, such as maxima, minima, or points of inflection.

In this case,
`\(x₀ = (5)/(4)\)` suggests a location where the function
`\(f(x)\)` may have a local extremum or an inflection point. Further analysis, such as the second derivative test, would be necessary to determine the nature of this critical point, whether it is a minimum, maximum, or an inflection point. Therefore, the critical value
`\(x₀\)` of the function is
`\((5)/(4)\).`

In calculus, critical values occur where the derivative is either zero or undefined. These points are potential locations of maxima, minima, or points of inflection on the graph of the function. In this case, the critical value
`\(x₀ = (5)/(4)\)` indicates a point where the function
`\(f(x)\)` may have a local extremum or an inflection point. To confirm the nature of this point, further analysis, such as the second derivative test, would be required.