#### Question

Let f(x) = 2x^{3}\[-\] 3x^{2}\[-\] 12x + 5 on [ 2, 4]. The relative maximum occurs at x = ______________ .

-2

-1

2

4

#### Solution

2

\[\text { Given }: f\left( x \right) = 2 x^3 - 3 x^2 - 12x + 5\]

\[ \Rightarrow f'\left( x \right) = 6 x^2 - 6x - 12\]

\[\text { For a local maxima or a local minima, we must have }\]

\[f'\left( x \right) = 0\]

\[ \Rightarrow 6 x^2 - 6x - 12 = 0\]

\[ \Rightarrow x^2 - x - 2 = 0\]

\[ \Rightarrow \left( x - 2 \right)\left( x + 1 \right) = 0\]

\[ \Rightarrow x = 2, - 1\]

\[\text{ Now, } \]

\[f''\left( x \right) = 12x - 6\]

\[ \Rightarrow f''\left( - 1 \right) = - 12 - 6 = - 18 < 0\]

\[\text { So, x = 1 is a local maxima } . \]

\[\text { Also }, \]

\[f''\left( 2 \right) = 24 - 6 = 18 > 0\]

\[\text { So, x = 2 is a local minima } . \]

Is there an error in this question or solution?

Solution Let F(X) = 2x3 − 3x2 − 12x + 5 on [ 2, 4]. the Relative Maximum Occurs at X = (A) − 2 (B) − 1 (C) 2 (D) 4 Concept: Graph of Maxima and Minima.