For all a, b in R,the function f(x) = 3x^4 - | vedclass

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(B) To find the extrema,we calculate the first derivative of $f(x)$:$f'(x) = 12x^3 - 12x^2 + 12x + a$Now,we find the second derivative to check the na

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exactly one extremum

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(B) To find the extrema,we calculate the first derivative of $f(x)$:$f'(x) = 12x^3 - 12x^2 + 12x + a$Now,we find the second derivative to check the nature of $f'(x)$:$f''(x) = 36x^2 - 24x + 12$We can rewrite $f''(x)$ as:$f''(x) = 12(3x^2 - 2x + 1)$The discriminant of the quadratic $3x^2 - 2x + 1$ is $D = (-2)^2 - 4(3)(1) = 4 - 12 = -8 Since the leading coefficient is positive and the discriminant is negative,$f''(x) > 0$ for all $x \in R$.This implies that $f'(x)$ is a strictly increasing function.$A$ strictly increasing polynomial of degree $3$ must cross the $x$-axis exactly once.Therefore,$f'(x) = 0$ has exactly one real root,which corresponds to exactly one extremum for the function $f(x)$.

For all $a, b \in R$,the function $f(x) = 3x^4 - 4x^3 + 6x^2 + ax + b$ has:

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