Let a and b be real constants such that the function f defined by f(x)= array|c|c| x^2+3x+…

Question

Let $a$ and $b$ be real constants such that the function $f$ defined by $f(x)= \begin{array}{|c|c|} {x}^{2}+3x+a & ,x\le 1 \ bx+2 & ,x>1 \end{array}$ be differentiable on $R$. Then, the value of $\int _{-2}^{2}f(x)dx$ equals

Accepted Answer

To determine the integral of the piecewise function $f$ over the interval $[-2,2]$, we first ensure that $f$ is differentiable on $R$, as given in the problem statement. Differentiability implies continuity, so $f$ must also be continuous at $x=1$.

The condition for continuity at $x=1$ is:

${x}^{2}+3x+a=bx+2$ at $x=1$.

This simplifies to:

$1+3+a=b(1)+2$

$\Rightarrow a=b-2$

The first derivative of $f$ gives us two different expressions depending on the value of $x$:

For $x\le 1$, ${f}^{'}(x)=2x+3$;
and for $x>1$, ${f}^{'}(x)=b$.

For $f$ to be differentiable at $x=1$, these derivatives must be equal at that point. Setting ${f}^{'}(1)$ from both expressions equal to each other gives $2(1)+3=b$, thus $b=5$. And from $a=b-2$, we have $a=3$.

Now, we can calculate the integral of $f$ over the specified interval:

$\int _{-2}^{1}({x}^{2}+3x+3)\, dx+\int _{1}^{2}(5x+2)\, dx$

$\begin{gathered} ={[\frac{{x}^{3}}{3}+\frac{3{x}^{2}}{2}+3x]}_{1}^{-2}+{[\frac{5{x}^{2}}{2}+2x]}_{2}^{1} \ =(\frac{1}{3}+\frac{3}{2}+3)-(\frac{-8}{3}+6-6)+(10+4-\frac{5}{2}-2) \ =6+\frac{3}{2}+12-\frac{5}{2}=17 \end{gathered}$

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