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Joint Entrance Examination
Mathematics
Application of Derivatives
Question 202
Question 202 of 212
Medium
MCQ
The function \[f(x)=2 x+3(x)^{\frac{2}{3}}, x \in \mathbb{R}\], has
A
exactly one point of local minima and no point of local maxima
B
exactly one point of local maxima and exactly one point of local minima
C
exactly two points of local maxima and exactly one point of local minima
D
exactly one point of local maxima and no point of local minima
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Official solution
\[
f
(
x
)
=
2
x
+
3
(
x
)
2
3
f
′
(
x
)
=
2
+
2
x
−
1
3
=
2
(
1
+
1
x
1
3
)
=
2
(
x
1
3
+
1
x
1
3
)
\begin{aligned} & f(x)=2 x+3(x)^{\frac{2}{3}} \\ & f^{\prime}(x)=2+2 x^{\frac{-1}{3}} \\ & =2\left(1+\frac{1}{x^{\frac{1}{3}}}\right) \\ & =2\left(\frac{x^{\frac{1}{3}}+1}{x^{\frac{1}{3}}}\right) \end{aligned}
f
(
x
)
=
2
x
+
3
(
x
)
3
2
f
′
(
x
)
=
2
+
2
x
3
−
1
=
2
(
1
+
x
3
1
1
)
=
2
(
x
3
1
x
3
1
+
1
)
\] So, maxima (M) at x = -1 \& minima (m) at x = 0
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