For which value(s) of a,f(x) = -x^3 + 4ax^2 + | vedclass

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(D) Given the function $f(x) = -x^3 + 4ax^2 + 2x - 5$. To find where the function is decreasing,we calculate the derivative $f'(x) = -3x^2 + 8ax + 2$.

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No value of $a$

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(D) Given the function $f(x) = -x^3 + 4ax^2 + 2x - 5$. To find where the function is decreasing,we calculate the derivative $f'(x) = -3x^2 + 8ax + 2$. For the function to be decreasing for every $x$,we must have $f'(x) This implies $-3x^2 + 8ax + 2 For a quadratic expression $Ax^2 + Bx + C$ to be negative for all $x$,the conditions are $A Here,$A = -3$,which is less than $0$. The discriminant is $\Delta = (8a)^2 - 4(-3)(2) = 64a^2 + 24$. Since $64a^2 + 24$ is always positive for any real value of $a$,the condition $\Delta Therefore,there is no value of $a$ for which the function is decreasing for every $x$.

For which value$(s)$ of $a$,$f(x) = -x^3 + 4ax^2 + 2x - 5$ is decreasing for every $x$?

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