Question

For x=4, the function f(x) = x^2 +bx+ c has the minimum value of -9 . Find c.

Answer 1

Answer:

$\bf \: f(x) = {x}^{2} + bx + c \\ \sf \: for \: x = 4 \: \: it \: has \: minmum \: value \\ \\ \therefore \: \sf \: x = - \frac{b}{2a} \\ \implies \: 4 = - \frac{b}{2 \times 1} \\ \bf \therefore \: b = - 8 \\ \\ \sf \: here \: coefficient \: of \: {x}^{2} \: is \: a = 1 > 0 \\ so \\ \\ \sf \: \: \: \: \: \: \: c - \frac{ {b}^{2} }{4a} = - 9 \\ = > \sf \: c = - 9 + \frac{ {b}^{2} }{4a} \\ \sf = > \: c = - 9 + \frac{ {( - 8)}^{2} }{4 \times 1} \\ \sf = > c = - 9 + \frac{64}{4} \\ \sf = > c = \frac{ - 36 + 64}{4} \\ \bf \therefore \:c = 7$

So c = 7

Note:

$\bf \: \: \: f(x) = a {x}^{2} + bx + c$

For maximum or minimum value of this polynomial

$\bf \: x = - \frac{b}{2a}$

if a>0 it has minimum value

if a<0 it has maximum value

And the value for both conditions is

$\sf \: c - \frac{ {b}^{2} }{4a}$

Answer 2

Answer:

$f(x)=x2+bx+cforx=4ithasminmumvalue∴x=−2ab⟹4=−2×1b∴b=−8herecoefficientofx2isa=1>0soc−4ab2=−9=>c=−9+4ab2=>c=−9+4×1(−8)2=>c=−9+464=>c=4−36+64∴c=7</p><p>So c = 7</p><p>Note:</p><p>\begin{gathered} \bf \: \: \: f(x) = a {x}^{2} + bx + c \\ \end{gathered}f(x)=ax2+bx+c</p><p>For maximum or minimum value of this polynomial</p><p>\bf \: x = - \frac{b}{2a}x=−2ab</p><p>if a>0 it has minimum value</p><p>if a<0 it has maximum value</p><p>And the value for both conditions is</p><p>\sf \: c - \frac{ {b}^{2} }{4a}c−4ab2</p><p>$