If f(x) is continuous and differentiable and $f (x) = a x^{4} + 5 x$ $f(x) = ax^4 + 5x$ ; $x \leq 2$ $x<=2$ and $b x^{2} - 3 x$ $bx^2 - 3x$ ; x> 2, then how do you find b?

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If f(x) is continuous and differentiable and $f (x) = a x^{4} + 5 x$ $f(x) = ax^4 + 5x$ ; $x \leq 2$ $x<=2$ and $b x^{2} - 3 x$ $bx^2 - 3x$ ; x> 2, then how do you find b?

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Answer 1 · 2016-11-07T08:32:08

$b = \frac{15}{2}$ $b = 15/2$

Explanation:

As $f (x)$ $f(x)$ is continuous at $x = 2$ $x=2$ , we have

$lim_(x->2^-)f(x) = lim_(x->2^+)f(x)$

$=> a(2^4)+5(2) = b(2^2)-3(2)$

$=> 16a+10 = 4b-6$

$=> a = 1/4b - 1$

As $f(x)$ is differentiable at $x=2$ , the limit $f'(2) = lim_(x->2)(f(x)-f(2))/(x-2)$ must exist. We can tell what the one sided limits will evaluate to by calculating the derivatives of the components of the piecewise defined functions on either side of $2$ .

$lim_(x->2^-)(f(x)-f(2))/(x-2) = lim_(x->2^+)(f(x)-f(2))/(x-2)$

$=> 4a(2^3)+5 = 2b(2)-3$

$=> 32a+5 = 4b-3$

Substituting in $a = 1/4b-1$ , we have

$32(1/4b-1) + 5 = 4b-3$

$=> 8b - 27 = 4b - 3$

$=> 4b = 30$

$:. b = 15/2$

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If f(x) is continuous and differentiable and $f (x) = a x^{4} + 5 x$ $f(x) = ax^4 + 5x$ ; $x \leq 2$ $x<=2$ and $b x^{2} - 3 x$ $bx^2 - 3x$ ; x> 2, then how do you find b?

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If f(x) is continuous and differentiable and f(x) = ax^4 + 5xf(x)=ax4+5xf(x) = ax^4 + 5x; x<=2x≤2x<=2 and bx^2 - 3xbx2−3xbx^2 - 3x; x> 2, then how do you find b?

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Explanation:

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If f(x) is continuous and differentiable and $f (x) = a x^{4} + 5 x$ $f(x) = ax^4 + 5x$ ; $x \leq 2$ $x<=2$ and $b x^{2} - 3 x$ $bx^2 - 3x$ ; x> 2, then how do you find b?