The set of real values of x- for which hx-1-2x2-4x4-6x6-100x100 is concave downward is

H(x)=1+2x^2+4x^4+6x^6+ ⋯+100x^100 h'(x)=0+2^2x+4^2x^3+6^2x^5+⋯+100^2x^99 h”(x)=2^2+4^2(3x^2)+6^2(5x^4)+⋯+100^2(99x^98) So, h”(x) is always positive for all real ...

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Standard XII

Mathematics

Sign of Quadratic Expression

The set of re...

Question

The set of real values of $x$ , for which $h (x) = 1 + 2 x^{2} + 4 x^{4} + 6 x^{6} + \dots + 100 x^{100}$ is concave downward is

x \in ϕ

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x \in W

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x \in R^{-}

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x \in R^{+}

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Solution

The correct option is A $x \in ϕ$
$h (x) = 1 + 2 x^{2} + 4 x^{4} + 6 x^{6} + \dots + 100 x^{100}$
$h^{'} (x) = 0 + 2^{2} x + 4^{2} x^{3} + 6^{2} x^{5} + \dots + 100^{2} x^{99}$
$h^{''} (x) = 2^{2} + 4^{2} (3 x^{2}) + 6^{2} (5 x^{4}) + \dots + 100^{2} (99 x^{98})$
So, $h^{''} (x)$ is always positive for all real values of $x$
Hence $h (x)$ is always concave upward

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The set of real values of x, for which h(x)=1+2x2+4x4+6x6+⋯+100x100 is concave downward is

The correct option is A x∈ϕh(x)=1+2x2+4x4+6x6+⋯+100x100 h′(x)=0+22x+42x3+62x5+⋯+1002x99 h′′(x)=22+42(3x2)+62(5x4)+⋯+1002(99x98) So, h′′(x) is always positive for all real values of x Hence h(x) is always concave upward

The set of real values of $x$ , for which $h (x) = 1 + 2 x^{2} + 4 x^{4} + 6 x^{6} + \dots + 100 x^{100}$ is concave downward is