Putting y = x we have,
$2f(x) + {\left[ {f(x)} \right]^2} = 1$
$2f(x) + {\left[ {f(x)} \right]^2} + 1 = 1 + 1$
${[f(x) + 1]^2} = 2$
$f(x) = – 1 \pm \sqrt 2 $
Since f(x) is a constant function, $f'(x) = 0$.
Category Archives: Math
$y = f(x) = \frac{x}{{x + 5}}$ $f(5x) = g(y)$ $g(y) = ?$
$f(5x) = \frac{{5x}}{{5x + 5}} = \frac{x}{{x + 1}}$
Given, $y = \frac{x}{{x + 5}}$
$\therefore y.x + 5y = x$ Continue reading $y = f(x) = \frac{x}{{x + 5}}$ $f(5x) = g(y)$ $g(y) = ?$
f'(0) = ? for an even function f that is differentiable at 0
Since f is differentiable at 0 we have,
$\mathop {\lim }\limits_{h \to 0} \frac{{f(0 + h) – f(0)}}{h} = \mathop {\lim }\limits_{h \to 0} \frac{{f(0 – h) – f(0)}}{{ – h}}$
$ \Rightarrow \mathop {\lim }\limits_{h \to 0} \frac{{f(h) – f(0)}}{h} = – \mathop {\lim }\limits_{h \to 0} \frac{{f( – h) – f(0)}}{h}$
Since f is an even function, f(-h) = f(h) Continue reading f'(0) = ? for an even function f that is differentiable at 0
$3{\cos ^2}x + 2\sin x + 1$
${(3{\cos ^2}x + 2\sin x + 1)_{\max }} = ?$
The given expression = $- 3{\sin ^2}x + 2\sin x + 4$
= – $\left[ {{{(\sqrt 3 \sin x)}^2} – 2.\frac{1}{{\sqrt 3 }}.\sqrt 3 \sin x + \frac{1}{3} – \frac{1}{3}} \right]$ + 4
$ = – \left[ {{{\left( {\sqrt 3 \sin x – \frac{1}{{\sqrt 3 }}} \right)}^2} – \frac{1}{3}} \right] + 4$ Continue reading $3{\cos ^2}x + 2\sin x + 1$
Solve for x & y$\sqrt x + y = 7$$x + \sqrt y = 11$
${x^{{x^2}}} = {x^{4x + 5}}$
The number of solution(s) the equation ${x^{{x^2}}} = {x^{4x + 5}}$ has:
(A) One solution only (B) Two solutions
(C) Three solutions (D) No solution
$\sin ({\pi ^x}) = {e^x} + {e^{ – x}}$Solve for real x
Using $A.M. \ge G.M.$,
$\frac{{{e^x} + {e^{ – x}}}}{2} \ge \sqrt {{e^x}.{e^{ – x}}} $
$ \Rightarrow {e^x} + {e^{ – x}} \ge 2$
$\therefore \sin ({\pi ^x}) \ge 2$ which is not possible for any real x.
So, no solution.