Question

Find both maximum and minimum vaule of 3x^4 -8x^3 +12x^2 -48x +25 on interval [0,3]​

Answer 1

$\large\underline{\sf{Solution-}}$

Given function is

$\rm :\longmapsto\:f(x) = {3x}^{4} - {8x}^{3} + {12x}^{2} - 48x + 25$

On differentiating both sides w. r. t. x, we get

$\rm :\longmapsto\:\dfrac{d}{dx}f(x) = \dfrac{d}{dx}( {3x}^{4} - {8x}^{3} + {12x}^{2} - 48x + 25 )$

$\rm :\longmapsto\:f'(x) = \dfrac{d}{dx}{3x}^{4} - \dfrac{d}{dx}{8x}^{3} + \dfrac{d}{dx}{12x}^{2} - \dfrac{d}{dx}48x + \dfrac{d}{dx}25$

$\rm :\longmapsto\:f'(x) = {12x}^{3} - {24x}^{2} + 24x - 48$

$\rm :\longmapsto\:f'(x) = {12x}^{2}(x - 2) + 24(x - 2)$

$\rm :\longmapsto\:f'(x) = ({12x}^{2} + 24)(x - 2)$

$\rm :\longmapsto\:f'(x) = 12({x}^{2} + 2)(x - 2)$

For maxima or minima,

$\boxed{ \tt{ \: f'(x) = 0 \: }}$

$\rm :\longmapsto\:12( {x}^{2} + 2)(x - 2) = 0$

$\bf\implies \:x = 2$

So, now calculate the value of f(x) at x = 0, 2, 3.

$\red{\rm :\longmapsto\:f(0) = {0}^{4} - {0}^{3} + {0}^{2} - 0 + 25}$

$\red{\bf\implies \:f(0) = 25}$

Now,

$\green{\rm :\longmapsto\:f(2) = {3(2)}^{4} - {8(2)}^{3} + {12(2)}^{2} - 48(2) + 25}$

$\green{\rm \: = \:48 - 64 + 48 - 96 + 25}$

$\green{\rm \: = - 39}$

$\red{\bf\implies \:f(2) = - 39}$

Now,

$\blue{\rm :\longmapsto\:f(3) = {3(3)}^{4} - {8(3)}^{3} + {12(3)}^{2} - 48(3) + 25}$

$\blue{\rm \: = \:243 - 216 + 108 - 144 + 25}$

$\blue{\rm \: = \:16}$

$\blue{\bf\implies \:f(3) = 16}$

So, of these, maximum value is f(0) = 25 and minimum value is f(2) = - 39.

Therefore,

$\red{\begin{gathered}\begin{gathered}\bf\: \rm :\longmapsto\:\begin{cases} &\sf{Maximum \: value = 25 \: at \: x = 0} \\ \\ &\sf{Minimum \: value = - 39 \: at \: x = 2} \end{cases}\end{gathered}\end{gathered}}$

More to know :-

$\green{\begin{gathered}\boxed{\begin{array}{c|c} \bf f(x) & \bf \dfrac{d}{dx}f(x) \\ \\ \frac{\qquad \qquad}{} & \frac{\qquad \qquad}{} \\ \sf k & \sf 0 \\ \\ \sf sinx & \sf cosx \\ \\ \sf cosx & \sf - \: sinx \\ \\ \sf tanx & \sf {sec}^{2}x \\ \\ \sf cotx & \sf - {cosec}^{2}x \\ \\ \sf secx & \sf secx \: tanx\\ \\ \sf cosecx & \sf - \: cosecx \: cotx\\ \\ \sf \sqrt{x} & \sf \dfrac{1}{2 \sqrt{x} } \\ \\ \sf logx & \sf \dfrac{1}{x}\\ \\ \sf {e}^{x} & \sf {e}^{x} \end{array}} \\ \end{gathered}}$